To solve the given recurrence relation, we use the formulas for the sum of the first n natural numbers and the sum of the squares of the first n natural numbers.
The given recurrence relation consists of two formulas:
∑i=1 i = n(n + 1) / 2 (Sum of the first n natural numbers)
∑i=1 i^2 = n(n + 1)(2n + 1) / 6 (Sum of the squares of the first n natural numbers)
These formulas are well-known and can be derived using various methods, such as mathematical induction or algebraic manipulation.
Using these formulas, we can substitute the given recurrence relation with the corresponding formulas to obtain an explicit solution.
For example, if we have a recurrence relation of the form ∑i=1 i^2 = 2∑i=1 i - 3, we can substitute the formulas to get:
n(n + 1)(2n + 1) / 6 = 2 * n(n + 1) / 2 - 3.
Simplifying the equation, we can solve for n and obtain the explicit solution to the recurrence relation.
In summary, to solve the given recurrence relation, we utilize the formulas for the sum of the first n natural numbers and the sum of the squares of the first n natural numbers. By substituting these formulas into the recurrence relation, we can simplify and solve for the unknown variable to obtain an explicit solution.
Learn more about recurrence relation: brainly.com/question/4082048
#SPJ11
To solve the given recurrence relation, we use the formulas for the sum of the first n natural numbers and the sum of the squares of the first n natural numbers.
The given recurrence relation consists of two formulas:
∑i=1 i = n(n + 1) / 2 (Sum of the first n natural numbers)
∑i=1 i^2 = n(n + 1)(2n + 1) / 6 (Sum of the squares of the first n natural numbers)
These formulas are well-known and can be derived using various methods, such as mathematical induction or algebraic manipulation.
Using these formulas, we can substitute the given recurrence relation with the corresponding formulas to obtain an explicit solution.
For example, if we have a recurrence relation of the form ∑i=1 i^2 = 2∑i=1 i - 3, we can substitute the formulas to get:
n(n + 1)(2n + 1) / 6 = 2 * n(n + 1) / 2 - 3.
Simplifying the equation, we can solve for n and obtain the explicit solution to the recurrence relation.
In summary, to solve the given recurrence relation, we utilize the formulas for the sum of the first n natural numbers and the sum of the squares of the first n natural numbers. By substituting these formulas into the recurrence relation, we can simplify and solve for the unknown variable to obtain an explicit solution.
Learn more about recurrence relation: brainly.com/question/4082048
#SPJ11
What is the minimum number N of integers that we can have so
that at least nine
have the same last digit?
The minimum number N of integers that we can have to ensure they all have the same last digit is 10.
To understand why, let's consider the possible last digits for numbers. There are 10 possible last digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
Now, if we have a set of N integers, each with a different last digit, we can conclude that N must be greater than or equal to 10. This is because if N is less than 10, at least two of the integers must have the same last digit.
For example, if we have only 9 integers, we can't have all 10 possible last digits represented. So, to ensure that all integers have the same last digit, we need a minimum of 10 integers.
For more similar questions on the same last digit
brainly.com/question/7989290
#SPJ8
Moving to the next question prevents changes to this answer. Question 8 Calculate the concentration of vibranium(IV) cation, Vb4+, in a saturated solution of VbCl4 (Ksp = 3,23x10-10) • Write your answer in scientific notation Example, 1.23x104 would be 1.23e-4 • Write you answer with 3 Significant figures • Show calculations in CALCULATIONS assignment Moving to the next question prevents changes to this answer. ㅇㅇ 박 novo
The concentration of Vb4+ in the saturated solution of VbCl4 is 3.23x10-10 mol/L.
To calculate the concentration of Vb4+, we need to use the solubility product constant (Ksp) equation. The balanced equation for the dissociation of VbCl4 is VbCl4 (s) ⇌ Vb4+ (aq) + 4Cl- (aq).
Since the concentration of Vb4+ is unknown, we can assign it a variable, let's say x. The concentration of Cl- is 4x (since there are 4 Cl- ions for every Vb4+ ion).
According to the Ksp expression, Ksp = [Vb4+][Cl-]^4. Plugging in the values, we have Ksp = x(4x)^4.
Now, we can solve for x by taking the fourth root of both sides and then substituting the value of Ksp: x = (Ksp)^(1/4).
x = (3.23x10-10)^(1/4) = 2.12x10-3 mol/L.
Therefore, the concentration of Vb4+ in the saturated solution of VbCl4 is 2.12x10-3 mol/L (or 2.12 mM).
Know more about concentration here:
https://brainly.com/question/30862855
#SPJ11
1. If (x−k) is a factor of x^4+2x^3−6x^2+8x−10 list all "possible values of k. (Do not solve). 2.Now consider the function p(x)=−5x^3+2x+6 List all the possible rational roots for this function. (Do not factor.)
1. The possible values of k are all the factors of the constant term of the polynomial divided by the leading coefficient.
2. The possible rational roots for the function p(x) = -5x^3 + 2x + 6 can be found by considering all the factors of the constant term divided by the leading coefficient.
For the first question, to find the possible values of k, we need to determine the factors of the constant term (-10) divided by the leading coefficient (1). In this case, the constant term is -10, so the factors of -10 are ±1, ±2, ±5, and ±10. Therefore, the possible values of k are 1, -1, 2, -2, 5, -5, 10, and -10.
Moving on to the second question, we are asked to find the possible rational roots of the function p(x) = -5x^3 + 2x + 6. To do this, we need to consider all the factors of the constant term (6) divided by the leading coefficient (-5). The constant term is 6, so the factors of 6 are ±1, ±2, ±3, and ±6. Dividing these factors by -5, we get the possible rational roots: -1/5, 1/5, -2/5, 2/5, -3/5, and 3/5.
Learn more about rational roots
brainly.com/question/29551180
#SPJ11
what is the range of the equation in the picture
The range of g(x) include the following: C. [-5, ∞).
What is a range?In Mathematics and Geometry, a range is the set of all real numbers that connects with the elements of a domain.
Based on the information provided about the piecewise-defined function, the range can be determined as follows:
g(x) = x² - 5, x < 2
g(x) = 0² - 5
g(x) = -5
g(x) = 2x, x ≥ 2
g(x) = 2(2)
g(x) = 4
Therefore, the range can be rewritten as -5 ≤ y ≤ ∞ or [-5, ∞].
Read more on range here: brainly.com/question/10684895
#SPJ1
Strontium-90 decays through the emission of beta particles. It has a half-life of 29 years. How long does it take for 80 percent of a sample of strontium-90 to decay? a) 21 years b) 9.3 years c) 38 years d) 96 years e) 67 years
The correct option is b) 9.3 years. It takes approximately 9.3 years for 80 percent of the sample of strontium-90 to decay.
To determine how long it takes for 80 percent of a sample of strontium-90 to decay, we can use the concept of half-life.
The half-life of strontium-90 is given as 29 years, which means that after 29 years, half of the original sample will have decayed.
If we want to find the time it takes for 80 percent of the sample to decay, we can calculate how many half-lives are required for this decay.
Let's denote the initial amount of strontium-90 as N0 and the remaining amount after time t as N.
Since each half-life corresponds to a 50 percent decay, we can write the equation:
N/N0 = (1/2)^(t/29)
To find the time t required for 80 percent of the sample to decay, we set N/N0 to 0.8 and solve for t:
0.8 = (1/2)^(t/29)
Taking the logarithm of both sides:
log(0.8) = log((1/2)^(t/29))
Using the logarithmic property, we can bring down the exponent:
log(0.8) = (t/29) log(1/2)
Solving for t:
t = (log(0.8) / log(1/2)) * 29
Calculating this expression:
t ≈ 9.3 years
Therefore, the correct option is b) 9.3 years. It takes approximately 9.3 years for 80 percent of the sample of strontium-90 to decay.
Learn more about decay from the given link
https://brainly.com/question/9932896
#SPJ11
1. Find the gross pay of an employee who worked 22 3/4 hours at an hourly rate of P18.00. 2. Patty received P618.75 gross pay for 33 hours worked. What is her hourly rate? 1. Determine the total hours worked by George if his hourly rate is P18.90 and his gross pay is P1,474.20. 2. Nancy works as a hairstylist. Her gross pay for last week was P407.00 and her hourly rate is P18.50. Calculate her total hours worked. 3. On Tuesday and Thursday, Margie worked 9 1/2 hours each day. Monday: Wednesday and Friday, she worked 7 hours each day. Her hourly rate is P20.00 plus time-and-a-half for any hours in excess of 8 per day. What is her gross pay? 4. Carol was paid P14.50 per hour with time-and-a-half for all hours worked in excess of 8 hours per day. She worked 9 ½ hours on Monday, 10 on Tuesday, 6 on Wednesday, 8 on Thursday and 11 on Friday. Find Carol's total pay for the week.
These calculations provide insights into the employee's earnings, hourly rates, and total hours worked, facilitating proper compensation and payroll management.
What is the gross pay for an employee who worked 22 3/4 hours at an hourly rate of P18.00?In the given scenarios, various calculations are performed to determine gross pay, hourly rate, or total hours worked.
The gross pay of an employee is calculated by multiplying the number of hours worked by the hourly rate.
To find the hourly rate, the gross pay is divided by the number of hours worked.
In some cases, the total hours worked are calculated by dividing the gross pay by the hourly rate.
Additional factors such as overtime or time-and-a-half rates are taken into account to calculate the gross pay accurately.
Learn more about compensation
brainly.com/question/28250225
#SPJ11
An air heater consists of a staggered tube bank in which waste hot water flows inside the tubes, with air flow through the bank perpendicular to the tubes. There are 30 rows of 15 mm-O.D. tubes, with transverse and longitudinal pitches of 28 and 32 mm, respectively. The air is at 1 atm and flows at 5.36 kg/s in a duct of 1.0 m square cross section. Preliminary design calculations for this heat exchanger suggest average tube surface and bulk air temperatures of approximately 350 K and 310 K, respectively. Estimate the average heat transfer coefficient and pressure drop across the bank.
The average heat transfer coefficient and pressure drop across the tube bank in the air heater, we can use empirical correlations.
1. Nu = 0.023 * (Re^0.8) * (Pr^0.4)
2. ΔP = (f * (L / D) * (ρ * V^2)) / 2
3. f = (0.79 * log(Re) - 1.64)^-2
1. Average Heat Transfer Coefficient (h):
The average heat transfer coefficient can be estimated using the Dittus-Boelter equation for forced convection:
Nu = 0.023 * (Re^0.8) * (Pr^0.4)
Where:
- Nu is the Nusselt number
- Re is the Reynolds number
- Pr is the Prandtl number
The Reynolds number (Re) can be calculated as:
Re = (ρ * V * D) / μ
Where:
- ρ is the density of air
- V is the velocity of air
- D is the hydraulic diameter of the tube (D = 4 * A / P, where A is the cross-sectional area and P is the wetted perimeter)
- μ is the dynamic viscosity of air
(Note: The values of ρ and μ can be obtained from air properties tables at the given bulk air temperature.)
The Prandtl number (Pr) can be approximated as:
Pr ≈ 0.7 (for air)
Once you calculate the Nusselt number (Nu), you can use it to determine the average heat transfer coefficient (h):
h = (Nu * k) / D
Where:
- k is the thermal conductivity of air
(Note: The value of k can be obtained from air properties tables at the given bulk air temperature.)
2. Pressure Drop (ΔP):
The pressure drop across the tube bank can be estimated using the Darcy-Weisbach equation:
ΔP = (f * (L / D) * (ρ * V^2)) / 2
Where:
- f is the friction factor
- L is the length of the flow path (number of rows * tube pitch)
- D is the hydraulic diameter of the tube
3. The friction factor (f) can be calculated using empirical correlations such as the Darcy friction factor equation for turbulent flow:
f = (0.79 * log(Re) - 1.64)^-2
Once you have the values of ΔP and V, you can calculate the pressure drop across the tube bank.
Remember to convert all units to the appropriate system (SI or consistent units) before performing the calculations.
Learn more about tube bank:
https://brainly.com/question/31495883
#SPJ11
The average heat transfer coefficient can be estimated using the Dittus-Boelter equation for forced convection: h ≈ XX [insert units] The pressure drop across the tube bank can be estimated using the Darcy-Weisbach equation: ΔP ≈ YY [insert units]
The average heat transfer coefficient and pressure drop across the tube bank in the air heater, we can use empirical correlations.
1. Nu = 0.023 * (Re^0.8) * (Pr^0.4)
2. ΔP = (f * (L / D) * (ρ * V^2)) / 2
3. f = (0.79 * log(Re) - 1.64)^-2
1. Average Heat Transfer Coefficient (h):
The average heat transfer coefficient can be estimated using the Dittus-Boelter equation for forced convection:
Nu = 0.023 * (Re^0.8) * (Pr^0.4)
Where:
- Nu is the Nusselt number
- Re is the Reynolds number
- Pr is the Prandtl number
The Reynolds number (Re) can be calculated as:
Re = (ρ * V * D) / μ
Where:
- ρ is the density of air
- V is the velocity of air
- D is the hydraulic diameter of the tube (D = 4 * A / P, where A is the cross-sectional area and P is the wetted perimeter)
- μ is the dynamic viscosity of air
(Note: The values of ρ and μ can be obtained from air properties tables at the given bulk air temperature.)
The Prandtl number (Pr) can be approximated as:
Pr ≈ 0.7 (for air)
Once you calculate the Nusselt number (Nu), you can use it to determine the average heat transfer coefficient (h):
h = (Nu * k) / D
Where:
- k is the thermal conductivity of air
(Note: The value of k can be obtained from air properties tables at the given bulk air temperature.)
2. Pressure Drop (ΔP):
The pressure drop across the tube bank can be estimated using the Darcy-Weisbach equation:
ΔP = (f * (L / D) * (ρ * V^2)) / 2
Where:
- f is the friction factor
- L is the length of the flow path (number of rows * tube pitch)
- D is the hydraulic diameter of the tube
3. The friction factor (f) can be calculated using empirical correlations such as the Darcy friction factor equation for turbulent flow:
f = (0.79 * log(Re) - 1.64)^-2
Once you have the values of ΔP and V, you can calculate the pressure drop across the tube bank.
Remember to convert all units to the appropriate system (SI or consistent units) before performing the calculations.
Learn more about tube bank:
brainly.com/question/31495883
#SPJ11
Question:
The standard curve for BSA can be used to assay proteins other than BSA. Why do you think this is possible? However, one protein for which the Coomassie dye is poor is collagen. Suggest a reason why this assay would not be appropriate.
The standard curve for BSA can be used to assay proteins other than BSA because the Coomassie dye, commonly used in protein assays, reacts with the peptide bonds in proteins in a relatively non-specific manner. The Coomassie dye used in protein assays may not effectively bind to or interact with these specific amino acid residues present in collagen.
The dye binds to the polypeptide backbone of proteins, resulting in a color change that can be measured spectrophotometrically. Since most proteins contain peptide bonds, the Coomassie dye can interact with and detect various proteins, allowing the standard curve for BSA to be used as a reference for protein quantification.
However, collagen is an exception to this general applicability of the assay. Collagen is a protein that has a unique structural composition, primarily consisting of repeating amino acid sequences rich in proline and hydroxyproline.
The Coomassie dye used in protein assays may not effectively bind to or interact with these specific amino acid residues present in collagen. As a result, the assay would not accurately detect or quantify collagen, leading to inaccurate results. Therefore, the Coomassie-based protein assay would not be appropriate for collagen analysis.
Learn more about protein at https://brainly.com/question/31357876
#SPJ11
Look at the picture below
Answer:
Step-by-step explanation:
By pythagoras theorem, a² + b² = hypotenuse²
1) √8 : 2 units and 2 units:
2² + 2² = 4 + 4 = 8 = (√8)²
2) √7 : √5 units and √2 units
(√5)² + (√2)² = 5 + 2 = 7 = (√7)²
3) √5 : 1 unit and 2 units
1² + 2² = 1 + 4 = 5 = (√5)²
4) 3 : 2 units and √5 units
2² + (√5)² = 4 + 5 = 9 = 3²
Elucidate the situation in which a disaster risk assessment report may recommend for the relocation of a development project to another area.
A disaster risk assessment report may recommend the relocation of a development project to another area in the following situation: When the current location is found to be at high risk or vulnerable to potential disasters.
A disaster risk assessment report evaluates the potential risks and vulnerabilities of a specific area or project to various hazards, such as natural disasters (e.g., earthquakes, floods, hurricanes), climate-related risks, or other significant threats. If the assessment determines that the current location of a development project poses a high level of risk or vulnerability to these hazards, it may recommend relocation to a safer area.
The primary reason for recommending the relocation of a development project based on a disaster risk assessment report is to mitigate the potential risks and vulnerabilities associated with the current location. By moving the project to an area with lower susceptibility to hazards, the report aims to reduce the potential impact of disasters and enhance the resilience of the project. Such a recommendation ensures the safety of the project, its occupants, and the surrounding community in the face of potential disasters.
Learn more about tential disasters visit:
https://brainly.com/question/19592780
#SPJ11
General Hospital's patient account division has compiled data on the age of accounts receivable The data collected indicate that the age of the accounts follows a normal distribution with meat 28 days and standard deviation 8 days. (i) What proportion of the accounts are between 25 and 40 days old? (ii) 60% of the accounts are aged above x days. Find the value of x. ( 2 marks)
The value of x for which 60% of the accounts are aged above x days is 30 days.The age of the accounts receivable data compiled by General Hospital's patient account division follows a normal distribution with a mean of 28 days and a standard deviation of 8 days.
The solutions to the given questions are given below:(i) The proportion of accounts that are between 25 and 40 days old can be calculated using the formula:
Z1 = (25 - 28) / 8 = - 0.375 and Z2 = (40 - 28) / 8 = 1.5
Now, using the z-table, the probability that corresponds to a z-score of -0.375 is 0.35 (approximately) and that corresponds to a z-score of 1.5 is 0.9332 (approximately).Thus, the proportion of the accounts that are between 25 and 40 days old is given by the difference between these probabilities:
P (25 < x < 40) = 0.9332 - 0.35= 0.5832
or approximately 58.32%
(ii) To find the value of x for which 60% of the accounts are aged above x days, we first need to find the z-score that corresponds to a probability of 0.6, using the z-table.
P(Z > z) = 0.6 or P(Z < z) = 0.4
Using the z-table, the z-score that corresponds to a probability of 0.4 is approximately 0.25.z = 0.25 Substituting the given values in the formula for z-score, we get:
z = (x - 28) / 8On solving for x, we get:x = 8z + 28= 8 × 0.25 + 28= 30
To know more about deviation visit:
https://brainly.com/question/14747159
#SPJ11
Explain how the following factors influence the recycling at
source:
Rural and urban communities
Developed and developing countries
Frequency of collection
Multi-dwelling and single dwelling houses
C
Factors like community type, country development, collection frequency, and housing type influence recycling at the source.
The factors mentioned have varying impacts on recycling at the source:
Rural and urban communities: Recycling in rural communities may be influenced by factors such as limited access to recycling facilities, fewer collection services, and lower awareness due to less exposure to recycling initiatives. In contrast, urban areas generally have more established recycling programs, better infrastructure, and higher awareness due to a larger population and greater exposure to recycling campaigns.Developed and developing countries: Developed countries often have well-established recycling systems with comprehensive collection services, recycling infrastructure, and strong government support. In developing countries, recycling at the source can be hindered by limited resources, inadequate infrastructure, and lower awareness. However, some developing countries are implementing initiatives to improve recycling practices.Frequency of collection: The frequency of collection significantly impacts recycling at the source. More frequent collections, such as weekly or bi-weekly, encourage residents to separate recyclables from waste and ensure timely disposal. Infrequent collections may lead to the accumulation of recyclables with regular waste, reducing the effectiveness of recycling efforts.Multi-dwelling and single dwelling houses: Recycling in multi-dwelling houses, such as apartment complexes, can be more challenging due to limited space for recycling bins and difficulties in implementing separate collection systems. In contrast, single dwelling houses typically have more space for recycling bins, making it easier to separate recyclables. However, effective education and infrastructure are essential for both types of dwellings to encourage recycling practices.In conclusion, factors such as community type, country development level, collection frequency, and housing type can influence recycling at the source. However, with the right infrastructure, education, and awareness campaigns, recycling can be promoted and improved in diverse settings.
Learn more about factors mentioned
brainly.com/question/1382896
#SPJ11
Whenever you see (aq) in equations for example in "CaCl2(s) → Ca2+ (aq) + 2 Cl - (aq)"
Should you assume water was used? Could it have been something else? Essentially if you see (aq) it should always assumed that water was used in any circumstance or depending on the equation/situation could it have been something else?
The (aq) in an equation indicates that the substance is dissolved in water or an aqueous solution. While water is commonly used as a solvent, it is not always the case. The choice of solvent depends on the specific circumstances and the nature of the reactants involved in the equation.
The (aq) in an equation stands for "aqueous," which means that the substance is dissolved in water. However, it is important to note that whenever you see (aq) in an equation, it doesn't necessarily mean that water was used as a reactant or a solvent.
In the given example equation "CaCl2(s) → Ca2+ (aq) + 2 Cl - (aq)", the (aq) represents that the calcium ions (Ca2+) and chloride ions (Cl-) are dissolved in water. It indicates that they are present in the aqueous phase after the reaction occurs.
In this circumstance, water is often used as a solvent because many ionic compounds, like calcium chloride (CaCl2), readily dissolve in water to form aqueous solutions. However, it is crucial to understand that the presence of (aq) doesn't always mean that water was used. It is possible for other solvents to be used in different equations or situations.
For example, in the reaction "NH4NO3(s) → NH4+ (aq) + NO3- (aq)", the (aq) represents that the ammonium ions (NH4+) and nitrate ions (NO3-) are dissolved in an aqueous solution. In this case, water is commonly used as the solvent, but it could also be another solvent suitable for dissolving the reactants.
To summarize, the (aq) in an equation indicates that the substance is dissolved in water or an aqueous solution. While water is commonly used as a solvent, it is not always the case. The choice of solvent depends on the specific circumstances and the nature of the reactants involved in the equation.
learn more about aqueous on :
https://brainly.com/question/19587902
#SPJ11
When you see (aq) in an equation, it typically implies that the substance is dissolved in water. However, in some cases, it can indicate a solute dissolved in a different solvent. It's important to consider the context and other information in the equation to determine the nature of the solvent.
When you see (aq) in an equation, it indicates that the substance is in an aqueous solution, meaning it is dissolved in water. However, it's important to note that not all aqueous solutions involve water. While water is the most common solvent, there are other substances that can also dissolve solutes and form aqueous solutions.
For example, in the equation "CaCl2(s) → Ca2+ (aq) + 2 Cl- (aq)," the (aq) indicates that calcium ions (Ca2+) and chloride ions (Cl-) are present in an aqueous solution. In this case, water is the most likely solvent. However, there are situations where other solvents can be used to form aqueous solutions. For instance, if the equation involves a non-water solvent, such as ethanol, the (aq) would indicate that the solute is dissolved in the specified solvent.
So, while (aq) generally suggests that water was used, it's not always the case. Depending on the specific equation or situation, (aq) can refer to a solute dissolved in a solvent other than water.
Learn more about aqueous solution from this link
https://brainly.com/question/19587902
#SPJ11
Q1 Discuss in steps the manufacturing of portland cement. Clarify the differences between dry and wet method. Q2 What is meant by hydraulic cement? State the basic chemical compounds of portland cement using the short hand notations.
The manufacturing process of Portland cement involves several steps. Here is a step-by-step explanation:
1. Quarrying: The process begins with the extraction of raw materials from limestone quarries. Limestone, clay, and other materials are typically used as raw materials.
2. Crushing and Grinding: The extracted raw materials are then crushed and ground into a fine powder. This step helps in increasing the surface area of the materials, allowing for better chemical reactions during the subsequent steps.
3. Mixing: The finely ground raw materials are mixed in the right proportions to form a homogeneous mixture. The typical composition of Portland cement includes around 60-65% limestone, 15-25% clay, 5-10% silica, and small amounts of other materials.
4. Heating: The mixture is then heated in a kiln at high temperatures (around 1450°C or 2640°F). This process, known as calcination, helps in removing the excess water and carbon dioxide, resulting in the formation of clinker.
5. Grinding the Clinker: The clinker is then ground into a fine powder along with a small amount of gypsum (calcium sulfate). This step helps in enhancing the setting properties of the cement.
Now, let's clarify the differences between the dry and wet methods of manufacturing Portland cement:
In the dry method, the raw materials are dried and ground separately. They are then mixed together and fed into the kiln. This method requires less energy and produces a lower-quality cement. In the wet method, the raw materials are mixed with water to form slurry before being fed into the kiln. This method is energy-intensive and produces a higher-quality cement.
Hydraulic cement is a type of cement that can set and harden even underwater. It is capable of developing strength through hydration reactions with water. Portland cement is a common type of hydraulic cement.
Now, let's discuss the basic chemical compounds of Portland cement using shorthand notations:
- C3S: Tricalcium silicate (3CaO·SiO2)
- C2S: Dicalcium silicate (2CaO·SiO2)
- C3A: Tricalcium aluminate (3CaO·Al2O3)
- C4AF: Tetracalcium aluminoferrite (4CaO·Al2O3·Fe2O3)
These compounds are responsible for the cement's setting and hardening properties. Tricalcium silicate (C3S) and dicalcium silicate (C2S) are the main compounds contributing to the strength development of Portland cement.
.Learn more about manufacturing process of Portland cement
https://brainly.in/question/54159396
#SPJ11
For 12C160 the lowest observed rotational absorption frequency is 115,271 x 106 s-1 a) the rotational constant? 12 b) length of the bond ¹2C¹6O
The rotational constant of ¹²C¹⁶O is 57,635.5 x 10^6 s⁻¹.
The bond length of ¹²C¹⁶O is approximately 1.128 x 10^(-10) meters.
To determine the rotational constant (B) and the bond length of ¹²C¹⁶O, we can use the formula for the rotational energy levels of a diatomic molecule:
E(J) = B * J(J+1)
where E(J) is the energy level corresponding to the rotational quantum number J, and B is the rotational constant.
a) Calculating the rotational constant (B):
Given the lowest observed rotational absorption frequency (ν) of 115,271 x 10^6 s⁻¹, we can use the formula:
ν = 2B
Rearranging the equation, we have:
B = ν/2
Substituting the given frequency, we get:
B = 115,271 x 10^6 s⁻¹ / 2 = 57,635.5 x 10^6 s⁻¹
b) Calculating the bond length (r):
The rotational constant (B) can be related to the moment of inertia (I) of the molecule by the following formula:
B = h / (8π²cI)
where h is Planck's constant, c is the speed of light, and I is the moment of inertia.
The moment of inertia (I) can be calculated using the reduced mass (μ) of the molecule and the bond length (r):
I = μr²
Rearranging the equation, we have:
r = √(I / μ)
To determine the reduced mass (μ) for ¹²C¹⁶O, we can use the atomic masses of carbon-12 (12.0000 g/mol) and oxygen-16 (15.9949 g/mol):
μ = (m₁m₂) / (m₁ + m₂)
μ = (12.0000 g/mol * 15.9949 g/mol) / (12.0000 g/mol + 15.9949 g/mol)
μ = 191.9728 g/mol
Now, we can calculate the bond length (r):
r = √(I / μ)
We need to determine the moment of inertia (I) using the rotational constant (B):
I = h / (8π²cB)
Substituting the known values into the equation:
I = (6.62607015 x 10^(-34) J·s) / (8π² * (2.998 x 10^8 m/s) * (57,635.5 x 10^6 s⁻¹))
I ≈ 2.789 x 10^(-46) kg·m²
Substituting the values of I and μ into the equation for r:
r = √(2.789 x 10^(-46) kg·m² / 191.9728 g/mol)
r ≈ 1.128 x 10^(-10) meters
Therefore, the bond length of ¹²C¹⁶O is approximately 1.128 x 10^(-10) meters.
learn more about rotational constant
https://brainly.com/question/26195587
#SPJ11
Determine the temperature of a reaction if K = 1.20 x 10-6 when AG° = +16.00 kJ/mol.
To convert kJ/mol to J/mol, multiply the given value by 1000:`AG° = 16.00 × 10³ J/mol T = 430.29 K. The temperature of a reaction if K = 1.20 × 10⁻⁶ when AG° = +16.00 kJ/mol is 157.14 °C approximately.
Let's convert the temperature in Kelvin to Celsius by subtracting 273.15:430.29 K - 273.15 = 157.14 °CSo.
The temperature of a reaction if K = 1.20 × 10⁻⁶ when AG° = +16.00 kJ/mol is given below;
According to the Gibbs-Helmholtz equation, the equilibrium constant K is related to the change in Gibbs free energy (AG°) of a reaction and the temperature (T) as follows:
`K = e^(-AG°/RT)`Where R is the universal gas constant (8.314 J K⁻¹ mol⁻¹), T is the temperature in Kelvin, and e is the mathematical constant (~ 2.718).
So, the temperature of a reaction if K = 1.20 × 10⁻⁶ when AG° = +16.00 kJ/mol is given as follows;`K = e^(-AG°/RT)`Let's rearrange this equation to solve for T:`lnK = -AG°/RT
Substitute the given values in the equation: AG° = +16.00 kJ/molK = 1.20 × 10⁻⁶R = 8.314 J K⁻¹ mol⁻¹
Substitute these values in the equation and solve for T:`ln(1.20 × 10⁻⁶) = -(16.00 × 10³)/(8.314 × T)`Solve for T:`T = -(16.00 × 10³)/(8.314 × ln(1.20 × 10⁻⁶))`T = 273.15 × (-(16.00 × 10³)/(8.314 × ln(1.20 × 10⁻⁶)))
Learn more about Kelvin:
https://brainly.com/question/30708681
#SPJ11
how
can geophysics survey methods be used in geometric road
designs
Geophysics survey methods aid in geometric road design by identifying soil layers with varying properties, such as strength, bearing capacity, compressibility, and deformation. This information helps engineers determine the best location, optimal design, and material requirements. Geophysical survey methods also help identify sinkholes and subsurface features, ensuring solid ground for road construction.
Geophysics survey methods are essential in geometric road designs, as they help identify soil layers with varying properties and strengths. These properties include soil strength, bearing capacity, compressibility, and deformation. Understanding these properties helps engineers determine the best location, optimal design, and material requirements for the road. Geophysics survey methods are particularly useful in locating buried utilities and identifying potential sinkholes, underground cavities, and other subsurface features that could affect road construction. This information is crucial for ensuring the road is built on solid ground that supports vehicle weight and withstands environmental factors.
The information obtained from geophysics survey methods can be used to create a subsurface map of the road site, which is then used to develop the best road design. Overall, geophysics survey methods are crucial in determining the properties of soil and subsurface features in geometric road designs, ultimately ensuring a safe and environmentally friendly road.
To know more about Geophysics survey methods Visit:
https://brainly.com/question/33113350
#SPJ11
3g of metal A density 2.7g/cm3 is mixed with 2.4dm3 of metal B of density 4.8g/cm3 determine the density of the mixture
Answer:
To determine the density of the mixture, we need to first find the total volume of the mixture, which can be calculated by adding the volumes of metal A and metal B.
The volume of metal A can be calculated using the formula:
Volume = Mass / Density
So, the volume of metal A is:
Volume of A = 3.3g / 2.7g/cm³ = 1.2222... cm³ (rounded to four decimal places)
Similarly, the volume of metal B is:
Volume of B = 2.4g / 4.8g/cm³ = 0.5 cm³
The total volume of the mixture is therefore:
Total Volume = Volume of A + Volume of B
= 1.2222... cm³ + 0.5 cm³
= 1.7222... cm³ (rounded to four decimal places)
To find the density of the mixture, we can use the formula:
Density = Mass / Volume
The total mass of the mixture is:
Total Mass = Mass of A + Mass of B
= 3.3g + 2.4g
= 5.7g
So, the density of the mixture is:
Density = Total Mass / Total Volume
= 5.7g / 1.7222... cm³
= 3.3103... g/cm³ (rounded to four decimal places)
Therefore, the density of the mixture is approximately 3.3103 g/cm³
Step-by-step explanation:
Hope this helps
The density of the mixture is 4.79903 g/cm³. To determine the density of a mixture, we must know the total mass and total volume of the mixture, and then we divide the total mass by the total volume.
Here, the mass and density of metal A are 3g and 2.7g/cm³ whereas, the volume and density of metal B are 2400cm³ and 4.8g/cm³ respectively. So, we need to find the volume of metal A and as for metal B, we need to find its mass. We know that the formula for finding density is:
Density = Total mass / Total volume
Now,
For Metal A:
Mass = 3g
Density = 2.7g/cm³
⇒Volume = 3/2.7 = 1.11 cm³
For Metal B:
Volume = 2.4 dm³ = 2400cm³
Density = 4.8g/cm³
⇒ Mass = 2400×4.8 = 11520g
Now, put the values in the equation,
Density = Total mass / Total volume
= (3+11520) / (1.11+2400)
Density= 4.79903 g/cm³
Thus, the density of the mixture is 4.79903 g/cm³.
Learn more about density from the below link:
https://brainly.com/question/28348989
Nick has £1200.
He pays £449 for a new TV.
His mortgage payment is £630.
How much money does he have left after paying for the TV and
paying his mortgage?
To calculate how much money Nick has left after paying for the TV and his mortgage, we need to subtract the total expenses from his initial amount.
Total expenses = TV payment + Mortgage payment
Total expenses = £449 + £630
Total expenses = £1079
Money left = Initial amount - Total expenses
Money left = £1200 - £1079
Money left = £121
Therefore, Nick has £121 left after paying for the TV and his mortgage.
Hopes this helps you out :D
Let T(x) and T(y) be the complete future lifetimes for the lives x and yrespectively. If T(x) and T(y) are independent show that: μxy=μx+μy
When T(x) and T(y) are independent, the mean of the joint future lifetimes is equal to the sum of the means of the individual future lifetimes.
If T(x) and T(y) are independent, it means that the lifetimes of individuals x and y are not related or influenced by each other. To show that μxy = μx + μy, where μxy represents the mean of the joint future lifetimes of x and y, and μx and μy represent the means of the future lifetimes of x and y respectively, we need to use the properties of independent random variables.
The mean of a random variable is also known as the expected value. In this case, we can express the mean of the joint future lifetimes as the sum of the means of the individual future lifetimes:
μxy = E[T(x) + T(y)]
Since T(x) and T(y) are independent, we can rewrite this expression as:
μxy = E[T(x)] + E[T(y)]
This equation shows that the mean of the joint future lifetimes is equal to the sum of the means of the individual future lifetimes, which is μx + μy. Therefore, μxy = μx + μy when T(x) and T(y) are independent.
Learn more about joint future from :
https://brainly.com/question/1270710
#SPJ11
Which equation shows the variable terms isolated on one side and the constant terms isolated on the other side for the equation -1/2x + 3 = 4 - 1/4x?
Answer:
x = -4
Step-by-step explanation:
To isolate the variable terms on one side and the constant terms on the other side of the equation -1/2x + 3 = 4 - 1/4x, we can follow these steps:
Move the constant term "3" to the right side of the equation by subtracting 3 from both sides:
-1/2x + 3 - 3 = 4 - 1/4x - 3
-1/2x = 1 - 1/4x
Combine like terms on each side of the equation:
-1/2x + 0 = 1 - 1/4x
Move the variable term "-1/4x" to the left side of the equation by adding 1/4x to both sides:
-1/2x + 1/4x = 1 - 1/4x + 1/4x
(-1/2 + 1/4)x = 1
Simplify the coefficients on the left side:
(-2/4 + 1/4)x = 1
(-1/4)x = 1
Multiply both sides of the equation by the reciprocal of -1/4, which is -4:
-4 * (-1/4)x = 1 * (-4)
x = -4
Therefore, the equation with the variable terms isolated on one side and the constant terms isolated on the other side is x = -4.
16. In a library the ratio of English books to Math books, is the same as the ratio of Math books to Science book. If there are 1200 books on English and 1800 books on Math, find the number of Science books.
17. Set up all the possible proportions from the numbers 12, 15, 8, 10.
18. Find the first term, if second, third and fourth terms are 21, 80, 120.
19. Find the second term, if first, third and fourth terms are 15, 27, 63.
20. Find the mean term, if the other two terms of a continued proportion are 15 and 60.
Answers for practice test on ratio and proportion are given below to check the exact answers of the questions.
The second term is 40.20. Let the mean term be x.Given, the two terms are 15 and 60.
Hence, x² = 15 × 60 ⇒ x = 30
Therefore, the mean term is 30.
16. Let the number of science books be x.
Therefore, the ratio of English books to Math books
= 1200/1800
= 2/3
The ratio of Math books to Science books
= 1800/x
Equating the two ratios,
we get:2/3
= 1800/x ⇒ x
= 2700
Thus, the number of Science books is 2700.17.
The four given numbers are 12, 15, 8, 10.
The possible proportions are:
12:15
= 4:512:8
= 3:212:10
= 6:515:8
= 15:815:10
= 3:220:8
= 5:220:10
= 2:118:10
= 9:5.18.
Let the first term be x.Common ratio, r
= (80/21)
= (120/80)
= (n/120) ⇒ n
= 180
Therefore, x
= 21/5
= 4.219.
Let the second term be x.Common ratio, r
= (27/15)
= (63/27)
= (81/x) ⇒ x
= 40.
The second term is 40.20. Let the mean term be x.Given, the two terms are 15 and 60.
Hence, x²
= 15 × 60 ⇒ x
= 30
Therefore, the mean term is 30.
To know more about term visit:
https://brainly.com/question/15387441
#SPJ11
Which of the following integrals represents the area of the surface obtained by rotating the curve y = e², 1≤ y ≤ 2, about the y-axis? + √² m (v) √/1 A. 2π 2 TT [ ²³ e² √/1 + (1/y)² dy e" B. 2TT C. 2T In(y) √/1 + (1/y)² dy D. 2TT 2 T²3√/1 + (1/y)² dy E. 2TT 2 ²9√/1 + (1/3) dy 2 [ ²³e³ √/1 + (1/3) dy 1 2 F. 2- /*In(y) √/1+ (1/3) dy 2
The integral that represents the area of the surface obtained by rotating the curve y = e², 1 ≤ y ≤ 2, about the y-axis is: A. 2π ∫[1,2] e² √(1 + (1/y)²) dy
To find the area of the surface obtained by rotating the curve y = e² about the y-axis, we can use the formula for the surface area of a solid of revolution.
The formula for the surface area of a solid of revolution, when the curve is rotated about the y-axis, is given by:
A = 2π ∫[a,b] f(y) √(1 + (f'(y))²) dy
In this case, the curve is y = e², and we want to find the area between y = 1 and y = 2. Therefore, the limits of integration are from 1 to 2.
Plugging in the given values, the integral becomes:
A = 2π ∫[1,2] e² √(1 + (1/y)²) dy
This represents the area of the surface obtained by rotating the curve y = e² about the y-axis between y = 1 and y = 2.
Note: The options B, C, D, E, and F do not correctly represent the integral for finding the surface area. Option B is simply 2π, which is not an integral and does not account for the shape of the curve. Options C, D, E, and F have incorrect integrands and limits of integration.
Learn more about area here: brainly.com/question/30307509
#SPJ11
2. Within the alkali metals (Group IA elements) does the distance of the valence electron from the nucleus increase or decrease as the atomic number increases? (Circle one) 3. Would the trend in atomic size that you described in question 2 cause an increase or a decrease in the attraction between the nucleus and the valence electron within the group as the atomic number increases? (Circle one)
The distance of the valence electron from the nucleus increases as the atomic number increases in the alkali metals (Group IA elements). As the atomic number of alkali metals (Group IA elements) increases, the distance between the valence electron and the nucleus increases, and the attraction between the nucleus and the valence electron decreases.
The alkali metals are situated in Group IA of the periodic table. The Group IA elements have one electron in their valence shell. The atomic size of the alkali metals increases from top to bottom within the group as the number of energy levels increases with the addition of electrons. As a result, the atomic radii increase down the group. Because the atomic number increases as you move down the group, so does the number of protons, which increases the positive charge of the nucleus.
However, the extra electron layer shields the positive charge of the nucleus, causing the valence electron to be farther away from the nucleus.3. As the atomic number increases within the group, the trend in atomic size would cause a decrease in the attraction between the nucleus and the valence electron. As we have learned, atomic size grows from top to bottom within the group as the valence electron moves away from the nucleus as the number of energy levels rises.
As a result, the attraction between the valence electron and the nucleus decreases as the valence electron moves further away from the nucleus. As the atomic number of alkali metals (Group IA elements) increases, the distance between the valence electron and the nucleus increases, and the attraction between the nucleus and the valence electron decreases.
To know more about periodic table, visit:
brainly.com/question/28747247
#SPJ11
What is the present value of a lottery paid as an annuity due for twenty years if the cash flows are $150,000 per year and the appropriate discount rate is 7.50%?
$5,000,000.00 $1,643.861.73 $2.739.769.55 $3,186,045.39
The present value of a lottery paid as an annuity due for twenty years if the cash flows are $150,000 per year and the appropriate discount rate is 7.50% is $1,643.861.73.
Calculation of the present value of a lottery paid as an annuity due for twenty years when the cash flows are $150,000 per year and the appropriate discount rate is 7.50% can be done using the formula:
PV = C * [(1 - (1 + r)^-n) / r] * (1 + r)
Where,C = Annual cash flow
r = Discount rate
n = Number of periods
PV = Present value
Given that,C = $150,000
r = 7.50%
n = 20
PV = $1,643,861.73
Therefore, the present value of a lottery paid as an annuity due for twenty years if the cash flows are $150,000 per year and the appropriate discount rate is 7.50% is $1,643.861.73.
To know more about annuity visit:
https://brainly.com/question/32931568
#SPJ11
One number is twelve iess than another number. The average of the two number is 96. What is the smaller of the two numbers? 92 90 102 a 84
Answer:
Smaller of the two numbers = 90
Step-by-step explanation:
We will need a system of equations to find the two numbers, where:
A represents one number,and B represents the other number.First equation:
Since one number is twelve less than the other number, our first equation is given by:
A = B - 12
Second equation:
The average of a set of numbers is the sum of the numbers divided by the amount of numbers in the set.Since there are two numbers and the average of the numbers is 96, our second equation is given by:
(A + B) / 2 = 96
Method to solve: Substitution:
We can solve for B by substituting A = B - 12 for A in (A + B) / 2 = 96.
(B - 12 + B) / 2 = 96
((2B - 12) / 2 = 96) * 2
(2B - 12 = 192) + 12
(2B = 204) / 2
B = 102
Thus, one of the numbers is 102.
Solving for A:
We can solve for A by plugging in 102 for B in A = B - 12:
A = 102 - 12
A = 90
Thus, the other number is 90.
Out of the two numbers, 90 is the smaller number.
Determine the heat transfer between two fluids separated by copper condenser tube of 20 mm outside dia, 1.8 m in length and wall thickness 2.5 mm if the outer (steam) temperature is 100 degree C and the inner (water) temperature is 15 degree C.. Assume that the water side film coefficient is 1400 kcal/m²-hr-deg and on the steam side is 9800 kcal/m²-hr-deg. Comment on the results.
The heat transfer rate between two fluids separated by a copper condenser tube is determined. The heat transfer rate is 5.72 kW.
Given data:Outer temperature, T1 = 100 °C
Inner temperature, T2 = 15 °C
Diameter of the copper tube,
D = 20 mm
= 0.02 m
Length of the copper tube, L = 1.8 m
Wall thickness of the copper tube,
δ = 2.5 mm
= 0.0025 m
Water side film coefficient, h1 = 1400 kcal/m²-hr-°C
Steam side film coefficient, h2 = 9800 kcal/m²-hr-°C
The heat transfer rate between two fluids separated by a copper condenser tube is given by,
Q = [pi * D * L / δ] * k * [ (T1 - T2) / (1/h1 + 1/h2 + pi * D * δ / k)]
where k is the thermal conductivity of copper.
= [3.14 × 0.02 × 1.8 / 0.0025] × 401 × [(100 - 15) / (1 / 1400 + 1 / 9800 + 3.14 × 0.02 × 0.0025 / 401)]
= 20600.32 kJ/hr
= 20600.32 / 3600
= 5.72 kW
This value is of great importance in condensers and heat exchangers. It is necessary to maintain an optimal heat transfer rate in the condenser and heat exchanger so that the desired heat transfer is achieved.
Know more about the heat transfer rate
https://brainly.com/question/33284608
#SPJ11
m = 10
mit 2. Solve the integration below (2 + m cos x) dx using Trapezoidal Method with a. n=10 b.n=15 c.n=40 Also, calculate the %error for each value of n. 5pts 5pts 5pts For this problem, let m be the 8t
To solve the integration ∫(2 + m cos x) dx using the Trapezoidal Method, we need to approximate the area under the curve by dividing it into smaller trapezoids.
Let's first substitute the given value of m into the expression: ∫(2 + 10 cos x) dx.
Using the Trapezoidal Method, we divide the interval of integration into smaller intervals.
a) For n = 10, we divide the interval into 10 smaller intervals. The width of each interval is Δx = (b - a) / n, where b and a are the limits of integration. Calculate the sum of the function values at the endpoints and the midpoints of each interval. Then, multiply the sum by Δx/2 to obtain the approximate area.
b) For n = 15, follow the same steps as in (a) but with 15 intervals.
c) For n = 40, repeat the process with 40 intervals.
To calculate the %error for each value of n, compare the approximated values to the exact solution. The %error is given by
[tex]|(exact - approximate)/exact| * 100.[/tex]
Remember to substitute the value of m back into the expression when calculating each integral.
Know more about Trapezoidal Method
https://brainly.com/question/32462209
#SPJ11
r
This table gives a few (x, y) pairs of a line in the coordinate plane.
x
Y
-12 14
-2
21
8 28
What is the x-intercept of the line?
Stuck? Review related articles/videos or use a hint.
Report a proble
The x-intercept of the line cannot be determined with the given information as there is no point in the table where the y-coordinate is zero.
To find the x-intercept of a line, we need to determine the value of x when y equals zero.
In other words, we are looking for the x-coordinate where the line intersects the x-axis.
Given the table of (x, y) pairs, we can observe that one of the pairs is (-2, 21).
However, this point does not lie on the x-axis, as the y-value is not zero.
Let's examine the other pairs:
(-12, 14)
(8, 28)
Since we are looking for the x-intercept, we need to find the point where y equals zero.
None of the given points satisfy this condition.
Based on the information provided, we do not have sufficient data to determine the x-intercept of the line.
Without any points where y equals zero, we cannot pinpoint the exact x-coordinate where the line intersects the x-axis.
It's important to note that the x-intercept represents the point(s) where a line crosses the x-axis.
If we had a point where y equals zero, we could determine the x-coordinate at that point.
However, in this case, the information given does not allow us to identify the x-intercept.
For similar question on x-intercept.
https://brainly.com/question/27921561
#SPJ8
You desire a cold, refreshing glass of water. You grab 20.0 g of ice at -7.2 °C. You add your ice to a thermos with 85.0 mL of water at 21.7 °C and wait until thermal equilibrium is established. Write your answers in the blanks provided. Show your work below. a) How much ice is present at thermal equilibrium? 5 grams b) What is the final temperature of the system? °C ice asystem = -asen 10
a. The mass of ice present at thermal equilibrium is mass of ice = 20.0 g * (T₃ - 21.7 °C) / 41.84 = 5 g.
b. The final temperature of the system is 22.6 °C
Determining the ice present at equilibriumTo solve this problem, use the principle of conservation of energy
The energy in the system is given by
E = E₁ + E₂
where E₁ is the thermal energy of the water and E₂ is the thermal energy of the ice.
When at thermal equilibrium, the final temperature of the system is the same throughout
E₁ + E₂ = E₃
where E₃ is the total thermal energy of the system at equilibrium.
The thermal energy of the water is given by
E₁ = mass of water * specific heat capacity of water * ΔTw
where ΔTw is the temperature change of the water. Since the water is at 21.7 °C initially and we assume it reaches thermal equilibrium with the ice, ΔT is the difference between the final temperature and the initial temperature:
ΔT = T₃ - 21.7
where T₃ is the final temperature of the system.
The thermal energy of the ice is given by:
E₂ = mass of the ice * specific heat capacity of ice* ΔTI
where ΔTI is the temperature change of the ice.
Since the ice is initially at -7.2 °C and we assume it reaches thermal equilibrium with the water, ΔTI is the difference between the final temperature and the initial temperature of the ice:
ΔTI = T₃ - (-7.2)
Now we can substitute these expressions for E₁ and E₂ into the conservation of energy equation and solve for the final temperature:
mass of water * specific heat capacity of water * (T₃- 21.7) + mass of ice * specific heat capacity of ice * (T₃+ 7.2) = mass of water * specific heat capacity of water * T₃ + mass of ice * L_f
where L_f is the latent heat of fusion of water (the amount of energy required to melt one gram of ice at 0 °C).
All of the ice will melt at thermal equilibrium, so we can solve for the mass of ice present at equilibrium by setting the right-hand side of the equation equal to zero
mass of ice * L_f = -mass of water * specific heat capacity of water * (T₃ - 21.7)
mass of ice = mass of water * specific heat capacity of water * (T₃ - 21.7) / L_f
Substitute the given values
mass of ice = 85.0 g * 4.18 J/(g·K) * (T₃ - 21.7 °C) / (333.5 J/g)
mass of ice = 20.0 g * (T₃- 21.7 °C) / 41.84
To find the final temperature, we can substitute this expression for mass of ice into the conservation of energy equation and solve for T₃:
85.0 g * 4.18 J/(g·K) * (T₃ - 21.7 °C) + 20.0 g * 2.09 J/(g·K) * (T₃ + 7.2 °C) = 0
355.3 T₃ - 8033.6 = 0
T₃ = 8033.6/355.3
= 22.6 °C
Therefore, the final temperature of the system is 22.6 °C, and the mass of ice present at thermal equilibrium is mass of ice = 20.0 g * (T₃ - 21.7 °C) / 41.84 = 5 g.
Learn more on thermal equilibrium on https://brainly.com/question/2642609
#SPJ4