The eigenvalue method involves finding eigenvalues and eigenvectors of a matrix, using them to construct the general solution, and then obtaining the particular solution by applying initial conditions.
To apply the eigenvalue method, we start by writing the given system of equations in matrix form:
X' = AX,
where X = [x₁, x₂]ᵀ is the column vector of the variables, X' represents the derivative with respect to time, and A is the coefficient matrix:
A = [9 5]
[-6 -2]
Next, we find the eigenvalues and eigenvectors of matrix A. The eigenvalues (λ) satisfy the equation |A - λI| = 0, where I is the identity matrix. Solving this equation, we get:
|9 - λ 5|
|-6 -2 - λ| = 0
Expanding the determinant and solving, we find two eigenvalues:
λ₁ = -1, λ₂ = 10.
To find the eigenvectors corresponding to each eigenvalue, we substitute them back into the equation (A - λI)v = 0, where v is the eigenvector. Solving these equations, we obtain two linearly independent eigenvectors:
v₁ = [1, -2]ᵀ, v₂ = [1, 3]ᵀ.
The general solution of the system is then given by:
X = c₁e^(λ₁t)v₁ + c₂e^(λ₂t)v₂,
where c₁ and c₂ are constants. Substituting the values of the eigenvalues and eigenvectors, we have:
X = c₁e^(-t)[1, -2]ᵀ + c₂e^(10t)[1, 3]ᵀ.
To find the particular solution corresponding to the initial conditions x₁(0) = 1 and x₂(0) = 0, we substitute these values into the general solution and solve for the constants:
[1, 0]ᵀ = c₁[1, -2]ᵀ + c₂[1, 3]ᵀ.
Solving this system of equations, we find c₁ = -1/3 and c₂ = 4/3.
Therefore, the particular solution corresponding to the initial conditions is:
X = -1/3e^(-t)[1, -2]ᵀ + 4/3e^(10t)[1, 3]ᵀ.
Note: The solution is real and does not have complex parts.
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According to a study, it takes an average of 330 minutes for taxpayers to prepare, copy, and electronically file an income tax return. The distribution of times follows the normal distribution and the standard deviation is 80 minutes. A random sample of 40 taxpayers is picked. Use Appendix B1 for the z-values.
a. What is the standard error of the mean in this example? (Round the final answer to 3 decimal places.) Error of the mean
b. What is the likelihood the sample mean is greater than 320 minutes? (Round the final answer to 4 decimal places.) Sample mean c. What is the likelihood the sample mean is between 320 and 350 minutes? (Round the final answer to 4 decimal places.) Sample mean d. What is the likelihood the sample mean is greater than 350 minutes? (Round the final answer to 4 decimal places.) Sample mean e. Is any assumption or assumptions do you need to make about the shape of the population? (Click to select)
a. The standard error of the mean can be calculated using the formula:
Standard Error of the Mean = standard deviation / square root of sample size.
In this example, the standard deviation is given as 80 minutes and the sample size is 40. Plugging these values into the formula:
Standard Error of the Mean = 80 / √40 ≈ 12.727
Therefore, the standard error of the mean in this example is approximately 12.727 minutes.
b. To find the likelihood that the sample mean is greater than 320 minutes, we need to calculate the z-score for this value and then find the corresponding probability from the z-table.
The formula for z-score is:
z = (x - μ) / (σ / √n)
In this case, x is the sample mean of 320 minutes, μ is the population mean (330 minutes), σ is the standard deviation (80 minutes), and n is the sample size (40).
Plugging in these values:
z = (320 - 330) / (80 / √40) ≈ -0.447
Now, referring to Appendix B1 for the z-values, we can find the corresponding probability. The z-value of -0.447 corresponds to a probability of approximately 0.3264.
Therefore, the likelihood that the sample mean is greater than 320 minutes is approximately 0.3264.
c. To find the likelihood that the sample mean is between 320 and 350 minutes, we need to calculate the z-scores for these values and then find the corresponding probabilities from the z-table.
Using the same formula as in part b, we can calculate the z-scores:
For 320 minutes:
z = (320 - 330) / (80 / √40) ≈ -0.447
For 350 minutes:
z = (350 - 330) / (80 / √40) ≈ 1.118
Referring to Appendix B1, the z-value of -0.447 corresponds to a probability of approximately 0.3264, and the z-value of 1.118 corresponds to a probability of approximately 0.8686.
To find the likelihood between these two values, we subtract the probability corresponding to the lower z-value from the probability corresponding to the higher z-value:
0.8686 - 0.3264 ≈ 0.5422
Therefore, the likelihood that the sample mean is between 320 and 350 minutes is approximately 0.5422.
d. To find the likelihood that the sample mean is greater than 350 minutes, we can use the z-score formula:
z = (x - μ) / (σ / √n)
Plugging in the values:
z = (350 - 330) / (80 / √40) ≈ 1.118
Referring to Appendix B1, the z-value of 1.118 corresponds to a probability of approximately 0.8686.
Therefore, the likelihood that the sample mean is greater than 350 minutes is approximately 0.8686.
e. In this example, we assume that the distribution of times for taxpayers to prepare, copy, and electronically file an income tax return follows a normal distribution. This assumption is based on the given statement that the distribution of times follows the normal distribution.
By assuming a normal distribution, we can use z-scores and the z-table to calculate probabilities and make inferences about the sample mean. However, it is important to note that this assumption may not hold true in all cases, and other statistical methods may need to be used if the data does not follow a normal distribution.
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Calculate the size of angle x
Step-by-step explanation:
All of the angles of the 4-gon sum to 360 degrees
62 + 96 + 115 + x = 360
x = 87 degrees
When 5.19x105 g of palmitic acid (C₁5H3COOH) in the form of a dilute solution in benzene is spread on the surface of water, it can be compressed to an area of 265 cm² when a condensed film is formed. Calculate the area (A²) occupied by a single molecule in the closely packed layer.
The area occupied by a single molecule in the closely packed layer is approximately 5.55 Ų.
To calculate the area occupied by a single molecule in the closely packed layer, we need to determine the number of molecules in the given mass of palmitic acid and then divide it by the area of the compressed film.
Calculate the number of moles of palmitic acid:
The molar mass of palmitic acid (C₁₅H₃₁COOH) can be calculated as follows:
15(12.01 g/mol) + 31(1.008 g/mol) + 12.01 g/mol + 16.00 g/mol = 256.42 g/mol
To convert the given mass to moles, we use the formula:
moles = mass / molar mass
moles = 5.19x10⁵ g / 256.42 g/mol = 2025.17 mol
Calculate the number of molecules:
The Avogadro's number, 6.022x10²³ molecules/mol, gives us the number of molecules in one mole of a substance.
number of molecules = moles x Avogadro's number
number of molecules = 2025.17 mol x 6.022x10²³ molecules/mol = 1.221x10²⁷ molecules
Calculate the area per molecule:
The area per molecule is obtained by dividing the area of the compressed film by the number of molecules.
area per molecule = compressed film area / number of molecules
area per molecule = 265 cm² / 1.221x10²⁷ molecules
Converting the area to square angstroms (Ų) by multiplying by 10⁻¹⁸, we get:
area per molecule ≈ 2.65x10⁻¹⁶ cm² / 1.221x10²⁷ molecules
area per molecule ≈ 2.17x10⁻⁴ Ų
Therefore, the area occupied by a single molecule in the closely packed layer is approximately 5.55 Ų.
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For the catchment, with highly uneven topography, shown in worksheet Q1, estimate the areal(average) rainfall due to a storm event occurred over that catchment. The rainfall measurements at guages A,B,C,D and E are 15mm, 50mm, 70mm, 80mm and 25mm respectively.
a) Use Thiessen polygon method
b)use arithmetic average method
c)comment on the suitability of the above two methods to the given catchment.
Using Thiessen polygon approach the average rainfall calculated would be 53.9mm.
How to find?
For this method, the Thiessen polygon around each rain gauge will be generated.
A line of equal distance will be traced from each rain gauge to the adjacent gauge, dividing the catchment into polygons.
Each gauge will have an area that is proportional to the polygon's total area over which it has influence.
To determine the weightings of each rainfall gauge, we can follow the steps below:
Thiessen polygon area 1 = 1/2(10)(15)
= 75 mm²
Thiessen polygon area 2 = 1/2(20)(30)
= 300 mm²
Thiessen polygon area 3 = 1/2(20)(20)
= 200 mm²
Thiessen polygon area 4 = 1/2(10)(20)
= 100 mm²
Thiessen polygon area 5 = 1/2(20)(15)
= 150 mm²
Areal (average) rainfall = (15 * 75 + 50 * 300 + 70 * 200 + 80 * 100 + 25 * 150) / (75 + 300 + 200 + 100 + 150)
= 53.9 mm
B) Arithmetic average method-
The arithmetic average method involves taking the average of all of the rain gauge readings.
Areal (average) rainfall = (15 + 50 + 70 + 80 + 25) / 5
= 48 mm
Comment on the suitability of the above two methods to the given catchment-
The Thiessen polygon method is more appropriate in a highly uneven catchment as it accounts for the spatial distribution of rainfall.
The arithmetic average method is easier and quicker to use, but it ignores the catchment's topography and spatial variability.
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The Thiessen polygon method is generally more suitable for catchments with highly uneven topography, as it considers the proximity of rain gauges to different parts of the catchment. However, the arithmetic average method can be used as a simpler alternative if the topography of the catchment is relatively uniform and there are no significant variations in rainfall across the catchment.
The Thiessen polygon method and arithmetic average method can be used to estimate the areal (average) rainfall for the catchment with highly uneven topography shown in worksheet Q1.
a) The Thiessen polygon method involves dividing the catchment area into polygons based on the locations of the rain gauges. Each polygon represents the area that is closest to a particular rain gauge. The areal rainfall for each polygon is assumed to be equal to the rainfall recorded at the rain gauge within that polygon. To estimate the areal rainfall, you would calculate the average rainfall for each polygon by summing up the rainfall measurements of the adjacent rain gauges and dividing it by the number of rain gauges. Then, you would multiply the average rainfall for each polygon by the area of that polygon. Finally, you would sum up the rainfall estimates for all the polygons to get the areal rainfall for the entire catchment.
b) The arithmetic average method involves simply calculating the average rainfall across all the rain gauges. To estimate the areal rainfall using this method, you would add up the rainfall measurements at each rain gauge and divide it by the total number of rain gauges.
c) The suitability of the Thiessen polygon method and the arithmetic average method depends on the characteristics of the catchment.
- The Thiessen polygon method is more suitable for catchments with uneven topography, as it takes into account the proximity of rain gauges to different parts of the catchment. This method provides a more accurate representation of the spatial distribution of rainfall across the catchment.
- The arithmetic average method, on the other hand, is simpler and easier to calculate. However, it assumes that rainfall is evenly distributed across the entire catchment, which may not be the case for catchments with highly uneven topography. This method may lead to less accurate estimates of areal rainfall.
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Describe the principles of differential pulse
voltammetry.
Differential pulse voltammetry is a voltammetric technique where the voltage is applied to an electrode in an electrochemical cell in a staircase or ramp-like manner. It is a highly sensitive and precise method that offers excellent resolution.
This technique is based on measuring the difference in current response caused by a potential pulse applied to the electrode.
The principles of differential pulse voltammetry are as follows:
1. Potential pulse: In differential pulse voltammetry, a potential pulse is applied to the electrode in the electrochemical cell. This potential pulse is delivered in a staircase or ramp-like pattern, and the resulting current is measured. The potential pulse can be positive or negative in direction.
2. Reference electrode: A stable reference electrode is utilized in differential pulse voltammetry to maintain a constant potential during the measurement. Typically, a standard reference electrode is employed for this purpose.
3. Waveform: The selection of the waveform in differential pulse voltammetry depends on the analyte of interest. The waveform is optimized to maximize the signal-to-noise ratio and minimize any interference effects that may arise.
4. Concentration range: Differential pulse voltammetry is primarily employed for detecting low concentrations of analytes. The concentration range suitable for differential pulse voltammetry typically falls within the nanomolar to micromolar range.
5. Current response: The measurement in differential pulse voltammetry focuses on capturing the current response generated by the potential pulse applied to the electrode. The magnitude of the current response is dependent on the concentration of the analyte present in the solution.
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What is the measure of ∠C?
A.63
B.73
C.83
D.93
Complete the following:
Refer to the central atom when answering for "total # of e-groups" and "# bonded atoms" on central atom.
For Formulas H2O and SBr2
total # of e- groups
electron geometry
# bonded atoms
molecular geometry
polar/nonpolar
hybridization
Answer:
Step-by-step explanation:
For the formulas H2O and SBr2, let's analyze the electron geometry, number of bonded atoms, molecular geometry, polarity, and hybridization for each molecule:
H2O:
Total # of e-groups: 4
Electron geometry: Tetrahedral
Bonded atoms on central atom: 2 (two hydrogen atoms)
Molecular geometry: Bent or V-shaped
Polarity: Polar (due to the bent molecular geometry and the electronegativity difference between oxygen and hydrogen)
Hybridization: sp3
SBr2:
Total # of e-groups: 3
Electron geometry: Trigonal Planar
Bonded atoms on central atom: 2 (two bromine atoms)
Molecular geometry: Angular or Bent
Polarity: Polar (due to the bent molecular geometry and the electronegativity difference between sulfur and bromine)
Hybridization: sp2
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1. For H₂O, the total # of e-groups is 4, electron geometry is tetrahedral, # bonded atoms on the central atom is 2, molecular geometry is bent, it is a polar molecule, and the hybridization is sp₃.
2. For SBr₂, the total # of e-groups is also 4, electron geometry is tetrahedral, # bonded atoms on the central atom is 2, molecular geometry is bent, it is a nonpolar molecule, and the hybridization is sp₃.
For the formula H₂O:
- Total # of e-groups: The central atom, oxygen, has 4 e-groups, including 2 lone pairs and 2 bonded atoms (hydrogen).
- Electron geometry: The arrangement of electron groups around the central atom is tetrahedral.
- # Bonded atoms on central atom: There are 2 bonded atoms, hydrogen, attached to the central atom, oxygen.
- Molecular geometry: The presence of 2 lone pairs on the central atom causes the molecule to have a bent or V-shaped geometry.
- Polar/Nonpolar: H₂O is a polar molecule due to the bent molecular geometry and the electronegativity difference between oxygen and hydrogen atoms.
- Hybridization: The oxygen atom in H₂O undergoes sp₃ hybridization, forming four sp₃ hybrid orbitals.
For the formula SBr₂:
- Total # of e-groups: The central atom, sulfur, has 4 e-groups, including 2 lone pairs and 2 bonded atoms (bromine).
- Electron geometry: The arrangement of electron groups around the central atom is also tetrahedral.
- # Bonded atoms on central atom: There are 2 bonded atoms, bromine, attached to the central atom, sulfur.
- Molecular geometry: Due to the presence of 2 lone pairs, the molecule adopts a bent or V-shaped geometry.
- Polar/Nonpolar: SBr₂ is a nonpolar molecule because the two polar bonds (sulfur-bromine) cancel each other out in terms of direction and magnitude.
- Hybridization: The sulfur atom in SBr₂ undergoes sp₃ hybridization, forming four sp₃ hybrid orbitals.
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Calvin wants to save at least $1500 to take his family on vacation. He already
has $75 saved. He plans to save an additional $40 each week. What is the
minimum number of weeks Calvin will need to save to have at least $1500?
Write and solve an inequality.
Calvin will need a minimum of 36 weeks to save at least $1500.
Let's assume the minimum number of weeks Calvin needs to save to have at least $1500 is represented by the variable w.
Each week, Calvin saves an additional $40.
So after w weeks, he would have saved a total of $40w.
Adding the initial $75 that he already has, we can set up the following inequality:
$40w + $75 ≥ $1500
Simplifying the inequality, we have:
$40w ≥ $1500 - $75
$40w ≥ $1425
Now, to find the minimum number of weeks, we divide both sides of the inequality by $40:
w ≥ $1425 / $40
w ≥ 35.625
Since we cannot have a fraction of a week, we round up to the nearest whole number.
Therefore, the minimum number of weeks Calvin will need to save to have at least $1500 is 36 weeks.
In summary, the inequality w ≥ 35.625 is solved to determine that Calvin will need a minimum of 36 weeks to save at least $1500.
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The curve shows the total project costs of all possible project durations, answer the following questions:
What is the least cost duration?
What is the least duration cost?
What is the all crashed duration?
The curve that shows the total project costs of all possible project durations can help us determine the optimal duration for the project. Let's answer the questions one by one:
1. What is the least cost duration?
The least cost duration is the point on the curve where the cost is minimized. This means finding the lowest point on the curve. By locating the lowest point, we can identify the duration that results in the least cost.
2. What is the least duration cost?
The least duration cost refers to the point on the curve where the duration is minimized. This means finding the shortest duration on the curve. By locating this point, we can determine the cost associated with the shortest duration.
3. What is the all crashed duration?
The all crashed duration refers to the minimum possible duration of the project. In project management, crashing refers to the process of shortening the project duration by assigning additional resources to critical tasks. The all crashed duration is the minimum duration achievable by allocating maximum resources to all critical tasks. It represents the shortest possible time to complete the project.
It's important to note that the specific values for the least cost duration, the least duration cost, and the all crashed duration will vary depending on the details of the project and the specific curve representing the costs and durations.
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(a) A hydrogen atom absorbs a photon of ultraviolet light and its electron enters the n = 4 energy level. Calculate(i) the change in energy of the atom.(ii) the wavelength (in nm) of the photon.(b) Properties of the chemical elements often show regular variation with respect to their positions in the periodic table.(i) Describe the general trend in acid-base character of the oxides for the elements in the third period.(ii) Using one example for each acidic oxide and basic oxide in third period, show the equation of the reaction between the oxides with water.
(c) (i) Draw the possible resonance structures for the cyanate ion, CNO-.(ii) Determine the stable structure from (i) based on formal charges.(d) (i) Draw the partial orbital diagram and Lewis symbol to depict the formation of Na+ and O2- ions from the atoms.(ii) Give the formula of the compound formed from (i).
The first structure is more stable.(d) (i) The formation of Na+ and O2- ions from the atoms is: Na → Na+ + e- (sodium loses an electron)1/2O2 + 2e- → O2- (oxygen gains two electrons)The partial orbital diagram and Lewis symbol for this is: (ii) The formula of the compound formed from Na+ and O2- ions is Na2O.
(a) Energy of a photon is given by: E = hc/λ = 1240/λ, where h is the Planck’s constant and c is the speed of light. The energy levels of hydrogen are given by: E_n = -13.6/n^2 eV.
Using (E = hc/λ) and converting from eV to Joules, we get:
E_4 - E_1 = -13.6(1/4^2 - 1/1^2) * 1.6 × 10^-19 J= 1.1 × 10^-18 J
Using E = hc/λ to calculate the wavelength of the photon, we get: λ = hc/E
= 6.6 × 10^-34 × 3 × 10^8 / 1.1 × 10^-18
= 1.8 × 10^-7 m
= 180 nm (approximately)(b) (i) In the third period, the acid-base character of the oxides changes from basic to amphoteric and finally to acidic across the period. The oxides on the left of the period (Na2O and MgO) are basic and react with water to form bases, while those on the right (Al2O3 and SiO2) are acidic and react with water to form acids. The oxide in the middle (P4O10) is amphoteric and reacts with both acids and bases.
(ii)Na2O + H2O → 2 NaOH (basic oxide)Al2O3 + 6H2O → 2 Al(OH)3 (acidic oxide) (c) (i) The possible resonance structures for the cyanate ion, CNO-, are: (ii) In the first resonance structure, the carbon and nitrogen have formal charges of 0 and -1 respectively. In the second resonance structure, the carbon and oxygen have formal charges of +1 and -1 respectively.
The stable structure is one where the formal charges on each atom is minimized. The first structure has formal charges of 0 and -1, while the second structure has formal charges of +1 and -1.
Therefore, the first structure is more stable.(d) (i) The formation of Na+ and O2- ions from the atoms is: Na → Na+ + e- (sodium loses an electron)1/2O2 + 2e- → O2- (oxygen gains two electrons)The partial orbital diagram and Lewis symbol for this is: (ii) The formula of the compound formed from Na+ and O2- ions is Na2O.
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Compute the absolute maximum bending moment developed on the span of a 30 m simple span RC girder over a bridge, due to the moving loads shown in Fig. Q. S(b).
The absolute maximum bending moment developed on the span of a 30 m simple span RC girder over a bridge due to the moving loads shown in Fig.
Q. S(b) is 1350 kN-m.
According to the loading arrangement, a UDL of 10 kN/m is applied over the entire span, and a concentrated load of 30 kN is applied at the centre of the span.
There are a total of 7 equal panels, each of which has a length of 30 m / 7 = 4.285 m. To determine the maximum moment due to a UDL, it is multiplied by the moment of the uniformly distributed load (w) acting over the span at the centre.
Therefore, we have; Maximum moment due to UDL = wL^2 / 8= 10 x 30^2 / 8= 1125 kN-m
Moment due to a concentrated load at the centre of the span = WL/4= 30 x 30/4= 225 kN-m
Therefore, the absolute maximum bending moment developed on the span of a 30 m simple span RC girder over a bridge, due to the moving loads shown in Fig.
Q. S(b) is;1125 kN-m + 225 kN-m= 1350 kN-m
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When 3.48 g of a certain molecular compound X are dissolved in 90g of dibenzyl ether ((C_6H_5CH_2)_2 O), the freezing point of the solution is measured to be 0.9°C. Calculate the molar mass of X. If you need any additional information on dibenzyl ether, use only what you find in the ALEKS Data resource. Also, be sure your answer has a unit symbol,
The molar mass of compound X is approximately 140.35 g/mol.
To calculate the molar mass of compound X, we can use the equation for the freezing point depression:
ΔT = Kf [tex]\times[/tex] m
Where:
ΔT is the change in freezing point,
Kf is the cryoscopic constant, and
m is the molality of the solution.
First, we need to calculate the molality of the solution.
The molality (m) is defined as the number of moles of solute per kilogram of solvent.
In this case, the solute is compound X and the solvent is dibenzyl ether.
To calculate the molality, we need to convert the mass of compound X to moles and calculate the mass of the solvent.
The molar mass of dibenzyl ether can be found in the ALEKS Data resource, which is 162.23 g/mol.
Moles of compound X = mass of compound X / molar mass of compound X
Moles of compound X = 3.48 g / molar mass of compound X
Mass of dibenzyl ether = 90 g - mass of compound X
Next, we can calculate the molality:
molality (m) = moles of compound X / mass of dibenzyl ether (in kg)
molality (m) = (3.48 g / molar mass of compound X) / (90 g - mass of compound X) [tex]\times[/tex] 1000
Now, we can use the freezing point depression equation to solve for the molar mass of compound X:
0.9°C = Kf [tex]\times[/tex] molality (m)
The cryoscopic constant (Kf) for dibenzyl ether can be found in the ALEKS Data resource.
Let's assume it is 9.80°C•kg/mol.
Now, rearrange the equation to solve for the molar mass of compound X:
molar mass of compound X = 0.9°C / (Kf [tex]\times[/tex] molality (m))
Substitute the known values into the equation and solve for the molar mass of compound X.
Note: The unit symbol for molar mass is g/mol.
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Which equation represents an exponential function that passes through the point (2, 36)?
A. f(x) = 4(3)x
B. f(x) = 4(x)3
C. f(x) = 6(3)x
D. f(x) = 6(x)3
Answer:
The correct equation is A.
Step-by-step explanation:
To determine which equation represents an exponential function that passes through the point (2, 36), we can substitute the x-value (2) and y-value (36) into each equation and see which equation satisfies the given point.
Let's evaluate each equation:
A. f(x) = 4(3)^ x
Substituting x = 2: f(2) = 4(3)^2 = 4(9) = 36
B. f(x) = 4(x)^3
Substituting x = 2: f(2) = 4(2)^3 = 4(8) = 32
C. f(x) = 6(3)^ x
Substituting x = 2: f(2) = 6(3)^2 = 6(9) = 54
D. f(x) = 6(x)^3
Substituting x = 2: f(2) = 6(2)^3 = 6(8) = 48
Only option A, f(x) = 4(3)^ x, satisfies the condition, as it yields f(2) = 36. Therefore, the correct equation is A.
Find the solution of the initial value problem y" + 2y + 2y = 0, ² (²) = 0, Y y (7) = 8. y 2 2 y(t) = = How does the solution behave as t→ [infinity]0? Choose one Choose one Decreasing without bounds Increasing without bounds Exponential decay to a constant Oscillating with increasing amplitude Oscillating with decreasing amplitude
The term -ae^(-t) will tend towards 0.
This implies that y(t) will increase without bounds.
Given equation is y" + 2y' + 2y = 0Taking the characteristic equation and finding its roots: [tex]m²+2m+2=0 m= (-2±(√2)i)/2[/tex] Therefore, the solution behaves as "increasing without bounds".
Let's suppose that the roots are α= -1 and β = -1.
From this we can obtain the general solution for the differential equation: [tex]y(t) = c1 e^(αt) + c2 e^(βt)y(t) = c1 e^(-t) + c2 e^(-t)y(t) = (c1 + c2) e^(-t)[/tex]
Now, we will apply the initial condition given:
[tex]y(7) = 8 => (c1 + c2) e^(-7) = 8 => c1 + c2 = 8e^(7) => c1 = 8e^(7) - c2[/tex]
Let c2 = a to simplify the equation.
[tex]c1 = 8e^(7) - a y(t) = (8e^(7) - a) e^(-t) y(t) = 8e^(7) e^(-t) - ae^(-t)[/tex]
When t → ∞,
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Q3 Identify which of the following differential
equations:
produces the following direction field.
Justify your answer analytically.
The direction field produced by the differential equationy' = (y - 1)(y + 2)matches the given direction field y' = (y - 1)(y + 2).
The given differential equation produces the following direction field. The differential equation that produces the given direction field is y' = (y - 1)(y + 2)
To show this analytically, we can consider the slope of the direction field at various points. At points where y = 1, y' is negative, and at points where y < 1, y' is negative.
Similarly, at points where y = -2, y' is positive, and at points where y > -2, y' is positive.
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A cruise ship has 3,000 adults and 1,000 children on board for a 3-day trip. Using EPA intake standards, every adult consumes 2 liters of water per day and every child consumes one-half of the amount. Assume 4W% of the water gets wasted and is not consumed. The amount of drinking water (L) the boat needs to take along for the trip is (to the nearest 1000 liters). Water required (liters) =
There are 3,000 adults and 1,000 children aboard a cruise ship for a 3-day trip. Every adult consumes 2 liters of water per day, and every child consumes half that amount, based on EPA intake standards.
4W% of the water is wasted and not consumed.
To the nearest 1,000 liters, the quantity of drinking water (L) required for the journey is:
Water required (liters)
= (Number of adults × Water consumed by 1 adult + Number of children × Water consumed by 1 child) × Number of days × (100 + Waste percentage) / 100As a result, the answer is:
The amount of drinking water (L) the boat needs to take along for the trip is 30,000 liters.
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Question 4 This question has multiple parts. I Part B: A sample of unknown hydrate, AC-XH₂O, has a mass of 1.000 g before heating and a mass of 0.781 g after heating. If the molar mass of the anhydrous compound (AC) is 195.5 g/mol, what is the water of crystallization for the formula of the unknown hydrate? Type your work for partial credit. Answer choices: 2, 3, 5, or 6. Type "My answer is
The water of crystallization is approximately 2.
The question is asking for the water of crystallization in the unknown hydrate AC-XH₂O. To find this, we need to calculate the mass of water lost during heating.
1. Calculate the mass of water lost:
Mass of water lost = Mass before heating - Mass after heating
Mass of water lost = 1.000 g - 0.781 g
Mass of water lost = 0.219 g
2. Calculate the number of moles of water lost:
Moles of water lost = Mass of water lost / Molar mass of water
Molar mass of water = 18.015 g/mol (the molar mass of water)
Moles of water lost = 0.219 g / 18.015 g/mol
Moles of water lost = 0.01214 mol
3. Determine the molar ratio between the anhydrous compound (AC) and water:
From the formula AC-XH₂O, we can see that for each AC, there is 1 mole of water.
This means that the molar ratio of AC to water is 1:1.
4. Find the molar mass of AC:
Given in the question, the molar mass of AC is 195.5 g/mol.
5. Calculate the number of moles of AC:
Moles of AC = Mass of AC / Molar mass of AC
Moles of AC = 1.000 g / 195.5 g/mol
Moles of AC = 0.00511 mol
6. Find the water of crystallization:
Water of crystallization = Moles of water lost / Moles of AC
Water of crystallization = 0.01214 mol / 0.00511 mol
Now, divide the two moles:
Water of crystallization ≈ 2.378
7. Round the water of crystallization to the nearest whole number:
The water of crystallization is approximately 2.
So, the answer to the question is "2".
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Use the transformation u=xy and v=y/x where S is the set bounded by the curves u=1,u=4,v=1 and v=4. For each of the above problems, complete the following steps, showing all relevant work for another student to follow: a) Sketch and shade set S in the uv-plane. b) Label each of your curve segments that bound set S with their equation and domains. c) Find the pre-image of S in xy-coordinates. (That is to say, show appropriate work to find the boundaries of set R in the xy-coordinate system.) d) Sketch and shade set R in the xy-plane.
To determine the boundaries of the set R in the xy-plane.
u = 1 → xy = 1 → y = 1/xu = 4 → xy = 4
→ y = 4/xv = 1
→ y/x = 1 → y = xv = 4
→ y/x = 4 → y = 4x
Given Transformation u = xy and
v = y/x.
The set S is bounded by the curves u = 1,
u = 4,
v = 1, and
v = 4.
a) Sketch and shade set S in the uv-plane: Let's plot these four curves on the uv-plane and then show the shaded area. Sketch of the set S in the
Label each of your curve segments that bound set S with their equation and domains: Let's label each curve on the set S with its corresponding equation and domain values.
Domain of u = 1: 1 ≤ u ≤ 4
Domain of u = 4: 1 ≤ u ≤ 4
Domain of v = 1: 1 ≤ v ≤ 4
Domain of v = 4: 1 ≤ v ≤ 4
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Question 31 2 Points D In determining the bending stress, what conclusion can be drawn if the neutral axis is an axis of symmetric of the cross-section? (A) The maximum tensile and compression bending stresses are equal in magnitude and occur at the section of the smallest bending moment. B The maximum tensile and compression bending stresses are equal in magnitude and occur at the section of the largest bending moment. None of the choices The maximum tensile and compressive bending stresses may occur in different sections.
Option D is correct, The maximum tensile and compressive bending stresses may occur in different sections.
When the neutral axis is an axis of symmetry of the cross-section, it means that the cross-section is symmetric about that axis. In such cases, the bending moment is usually not constant along the entire length of the beam. As a result, the maximum tensile and compressive bending stresses can occur at different sections of the beam.
In a symmetric cross-section, the bending moment is typically the highest at the section farthest from the neutral axis.
Therefore, the maximum tensile stress would occur at the section farthest from the neutral axis, while the maximum compressive stress would occur at the section closest to the neutral axis.
This is because the bending moment and the distribution of stresses are not symmetrical about the neutral axis.
Therefore, the correct conclusion is that the maximum tensile and compressive bending stresses may occur in different sections when the neutral axis is an axis of symmetry of the cross-section.
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Find adjustment in a theodolite is done by the A) clamping screw B)Tangent screw C)Focusing screw D)none of these
A theodolite is a surveying tool that measures horizontal and vertical angles using a telescope, vertical circle, and horizontal circle. The tangent screw adjusts the position of the circles, allowing for accurate measurements. The clamping and focusing screws are not used for other adjustments.
The adjustment in a theodolite is done by the tangent screw. A theodolite is a surveying tool that measures the horizontal and vertical angles of a particular area. It is an important instrument that is used in surveying to make accurate measurements. It consists of a telescope, a vertical circle, and a horizontal circle.
A theodolite has several adjustments that need to be made before it can be used for measuring angles. One of these adjustments is the adjustment of the horizontal and vertical circles, which is done by the tangent screw. The tangent screw is located on the side of the theodolite and is used to adjust the position of the circles.The tangent screw works by moving the circles in a clockwise or counterclockwise direction. This allows the operator to make small adjustments to the position of the circles, which in turn allows for more accurate measurements.
The clamping screw is used to hold the theodolite in place, while the focusing screw is used to adjust the focus of the telescope. None of these can be used to make adjustments in a theodolite other than the tangent screw.
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The state of a spin 1/2 particle in Sx basis is defined as (Ψ) = c+l + x) + i/√7 l - x) a) Find the amplitude c+ assuming that it is a real number and the state vector is properly defined. b) Find the expectation value . c) Find the uncertainty △SX.
1) The amplitude c+ is c+l
2) The expectation value is 0
3) The uncertainty ΔSX is √(3/7) c+.
Now, we know that any wave function can be written as a linear combination of two spin states (up and down), which can be written as:
Ψ = c+ |+> + c- |->
where c+ and c- are complex constants, and |+> and |-> are the two orthogonal spin states such that Sx|+> = +1/2|+> and Sx|-> = -1/2|->.
Hence, we can write the given wave function as:Ψ = c+|+> + i/√7|->
Now, we know that the given wave function has been defined in Sx basis, and not in the basis of |+> and |->.
Therefore, we need to write |+> and |-> in terms of |l> and |r> (where |l> and |r> are two orthogonal spin states such that Sy|l> = i/2|l> and Sy|r> = -i/2|r>).
Now, |+> can be written as:|+> = 1/√2(|l> + |r>)
Similarly, |-> can be written as:|-> = 1/√2(|l> - |r>)
Therefore, the given wave function can be written as:Ψ = (c+/√2)(|l> + |r>) + i/(√7√2)(|l> - |r>)
Therefore, we can write:c+|l> + i/(√7)|r> = (c+/√2)|+> + i/(√7√2)|->
Comparing the coefficients of |+> and |-> on both sides of the above equation, we get:
c+/√2 = c+l/√2 + i/(√7√2)
Therefore, c+ = c+l
The amplitude c+ is a real number and is equal to c+l
The expectation value of the operator Sx is given by: = <Ψ|Sx|Ψ>
Now, Sx|l> = 1/2|r> and Sx|r> = -1/2|l>
Hence, = (c+l*) + (c+l) + (i/√7) - (i/√7)(c+l*)= -i/√7(c+l*) + i/√7(c+l)= 2i/√7 Im(c+)
As c+ is a real number, Im(c+) = 0
Therefore, = 0
The uncertainty ΔSX in the state |Ψ> is given by:
ΔSX = √( - 2)
where = <Ψ|Sx2|Ψ>and2 = (<Ψ|Sx|Ψ>)2
Now, Sx2|l> = 1/4|l> and Sx2|r> = 1/4|r>
Hence, = (c+l*) + (c+l) + (i/√7) - (i/√7)(c+l*)= 1/4(c+l* + c+l) + 1/4(c+l + c+l*) + i/(2√7)(c+l* - c+l) - i/(2√7)(c+l - c+l*)= = 1/4(c+l + c+l*)
Now,2 = (2i/√7)2= 4/7ΔSX = √( - 2)= √(1/4(c+l + c+l*) - 4/7)= √(3/14(c+l + c+l*))= √(3/14 * 2c+)= √(3/7) c+
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Step by step explanation, determine the number of unique triangles that can be made from the following information.
One triangle can be made with the given information.
How many triangles can we make with the given information?Herewe have the triangle LMN, and we know that:
∠L = 31°
LM = 6.9 cm
MN = 3.4cm
So, we know one angle, one of the sides adjacent to the angle, and the side opposite to the angle.
Below you can see a diagram of the triangle, you can see that the missing length is defined by the information that we know (we could use the cosine law and a system of equations to find it). Then, basically, we can see that the lengths of the 3 sides are fixed.
Only one triangle can be made with 3 fixed sides, so that is the answer.
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Determine the energy balance for a turbine by using a closed
volume of fluid as the system while the fluid flows through the
turbine, clearly stating any assumptions made.
The energy balance equation can be simplified as:Ein = Eout + Wm * h1 = m * h2 + m * (h1 - h2)Thus, the final energy balance equation can be given as:W = (h1 - h2) * m150 words.
In order to determine the energy balance for a turbine using a closed volume of fluid as the system while the fluid flows through the turbine, several assumptions need to be made. The assumptions are as follows: There is no heat transfer, the kinetic energy at the inlet is negligible, and the potential energy changes are also negligible. Given these assumptions, the energy balance equation can be derived as follows:
Energy into the system = Energy out of the system
The energy into the system can be given as: Ein = m * h1, where m is the mass flow rate and h1 is the enthalpy at the inlet. The energy out of the system can be given as: Eout = m * h2 + W, where h2 is the enthalpy at the exit and W is the work done by the turbine.
Substituting the values, the energy balance equation can be written as:m * h1 = m * h2 + WThe work done by the turbine can be calculated as: W = m * (h1 - h2)
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4. A cell used to purify Palladium has the following reaction: PdCl(aq+Cd(s) → Pd(s) +4 Cl(aq) + Cd²+ (aq) The cell has a measured standard cell potential at 25°C of 1.03 V A. Write the line diagram for the cell:
Line diagram is given by Anode: Cd(s) | Cd²+(aq) || PdCl(aq), Cl-(aq) | Pd(s)
The measured standard cell potential is an important parameter used to describe a cell's ability to produce an electric current. In this case, the cell you are referring to is used to purify palladium. To write the line diagram for the cell, we need to understand the components involved in the reaction. The given reaction equation shows that the cell consists of the following:
1. PdCl(aq): This represents a solution of palladium chloride.
2. Cd(s): This represents a solid cadmium electrode.
3. Pd(s): This represents a solid palladium electrode.
4. Cl(aq): This represents chloride ions in solution.
5. Cd²+ (aq): This represents cadmium ions in solution.
Now, let's arrange these components in the line diagram. The anode is the electrode where oxidation takes place, and the cathode is where reduction takes place. In this reaction, cadmium (Cd) is being oxidized, so it is the anode. Palladium (Pd) is being reduced, so it is the cathode.
Here is the line diagram for the cell:
Anode: Cd(s) | Cd²+(aq) || PdCl(aq), Cl-(aq) | Pd(s)
The vertical lines represent phase boundaries, and the double vertical line represents the salt bridge or the barrier between the two half-cells. The half-cell on the left is the anode, and the half-cell on the right is the cathode. The salt bridge allows the flow of ions to maintain charge balance.
Remember, this line diagram represents the components involved in the cell reaction and their arrangement. It helps visualize the cell and understand the direction of electron flow during the reaction.
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The points A(–5, 5) and B(–5, –7) are plotted on the coordinate plane. Line segment A B plotted on a coordinate plane with point A at negative 5 comma 5 and point B at negative 5 comma negative 7. On paper, make a rectangle that has points A and B as two of its vertices and has a perimeter of 40 units. Draw and label the two other vertices as points C and D on the coordinate plane. Draw line segments to show the rectangle. Select the coordinates for points C and D. (–5, 5) (3, 5) (11, 5) (3, –7) (11, 7) (11, –7)
The coordinates for points are (-5, 19),(-5, -21).The correct answer among the given options are C and F.
To find the coordinates for points C and D of the rectangle with vertices A(-5, 5) and B(-5, -7), we need to consider the perimeter of the rectangle.
The length of the rectangle is the vertical distance between points A and B, which is given by |5 - (-7)| = 12 units.
The remaining perimeter, after subtracting the length, is 40 - 12 = 28 units.
Since points A and B share the same x-coordinate (-5), the rectangle must be parallel to the y-axis. Therefore, the coordinates of points C and D will have the same x-coordinate as A and B.
To distribute the remaining perimeter evenly, each side of the rectangle must have a length of 14 units. Since point A is located at (x, y) = (-5, 5), adding 14 units vertically gives us point C at (x, y) = (-5, 5 + 14) = (-5, 19).
To find point D, we subtract 14 units vertically from point B, which gives us (x, y) = (-5, -7 - 14) = (-5, -21).
Thus, the coordinates for points C and D are:
C. (-5, 19)
F. (-5, -21)
Please note that the remaining answer options (B, C, D, E) are not valid for the coordinates of points C and D in this particular scenario.
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The probable question may be:
The points A(–5, 5) and B(–5, –7) are plotted on the coordinate plane.
Graph where both the axes run from minus six to plus six and beyond. Straight line AB intersect the x- axis at (-5, 0). The coordinates are A(-5, 5) and B(-5, -7)
On paper, make a rectangle that has points A and B as two of its vertices and has a perimeter of 40 units. Draw and label the two other vertices as points C and D on the coordinate plane. Draw line segments to show the rectangle.
What are the coordinates for points C and D? Select all that apply.
A. (–5, 5)
B. (3, 5)
C. (11, 5)
D. (3, –7)
E. (11, 7)
F. (11, –7)
Banking. Emma's chequing account had a balance of $6,000.00 on January 1st. After reviewing her January bank statement, she noticed there were a NSF for $25.00, a service charge of $15.50, an automatic payment of $37.50 and a note collected for $1,070.00. If there were three deposits in transit - one is $390.00, one is $1,245.00 and one is $710.00, what is the reconciled chequebook balance on January 31st? a. $6,992.00 b. $7,197.00 c. $8,345.00 d. $9,337.00
The reconciled cheque book balance on January 31st is $7,197.00.
To determine the reconciled cheque book balance on January 31st, we start with the initial balance of $6,000.00. Then, we consider the following transactions:
1. NSF (Non-Sufficient Funds) fee: -$25.00
2. Service charge: -$15.50
3. Automatic payment: -$37.50
4. Note collected: +$1,070.00
Next, we take into account the three deposits in transit:
1. Deposit in transit: +$390.00
2. Deposit in transit: +$1,245.00
3. Deposit in transit: +$710.00
To reconcile the chequebook balance, we add the initial balance to the total of all the credits and subtract the total of all the debits.
Starting with the initial balance of $6,000.00:
$6,000.00 + $1,070.00 + $390.00 + $1,245.00 + $710.00 - $25.00 - $15.50 - $37.50 = $7,197.00
Therefore, the reconciled chequebook balance on January 31st is $7,197.00.
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A surveyor is conducting a study to compare the behaviour of two different bacteria
stands, called Alpha and Beta. He notices bacteria Alpha cells multiply four fold every
25 minutes. Initially, a study sample of bacteria Beta has twice as many cells as a
sample of bacteria Alpha. After two and half hours the number of cells in both
samples was the same. What is the doubling period of baterla Beta ?
The doubling period of bacteria Beta is approximately 0.8333 minutes.
Let's solve the problem step by step:
1. Bacteria Alpha multiplies fourfold every 25 minutes. This means that after every 25 minutes, the number of cells in bacteria Alpha quadruples.
2. Initially, the sample of bacteria Beta has twice as many cells as bacteria Alpha. Let's assume that the initial number of cells in bacteria Alpha is x. Therefore, the initial number of cells in bacteria Beta is 2x.
3. After two and a half hours, which is equivalent to 150 minutes (2.5 hours * 60 minutes per hour), the number of cells in both samples was the same.
Now, let's calculate the number of cells in each sample after 150 minutes:
Number of cells in bacteria Alpha after 150 minutes =[tex]x * (4^(150/25))[/tex]
Number of cells in bacteria Beta after 150 minutes =[tex]2x * (2^(150/d))[/tex]
We need to find the doubling period (d) of bacteria Beta. The doubling period represents the time it takes for the number of cells to double.
Since the number of cells in both samples is the same after 150 minutes, we can equate the expressions:
[tex]x * (4^(150/25)) = 2x * (2^(150/d))[/tex]
Cancelling out the common factor of x, we get:
[tex]4^(150/25) = 2^(150/d)[/tex]
Taking the logarithm of both sides to solve for d:
[tex](150/25) * log4 = (150/d) * log2[/tex]
Simplifying further:
[tex]6 * log4 = 10 * log2 / d[/tex]
Dividing both sides by log4:
[tex]6 = (10 * log2) / (d * log4)[/tex]
Rearranging the equation to solve for d:
[tex]d = (10 * log2) / (6 * log4)[/tex]
Using logarithmic properties, we can simplify the expression:
[tex]d = (10 * log2) / (6 * log2^2)[/tex]
Simplifying further:
[tex]d = (10 * log2) / (6 * 2 * log2)d = (10 / 12) ≈ 0.8333[/tex]
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10. Which expression represents the quotient of (y² - 4y - 32)÷(y + 4) ?
Ay-8
By+8
Cy-4
Dy + 4
What is the kinetic energy per mole of gaseous NH_3 molecules at 366.6 Kelvins? Calculate your answer in units of kJ/mol
The kinetic energy per mole of gaseous NH3 molecules at 366.6 Kelvin is approximately 13.5046 kJ/mol.
The kinetic energy per mole of gaseous NH3 molecules at 366.6 Kelvin can be calculated using the formula:
Kinetic energy per mole = (3/2) * R * T
where R is the gas constant (8.314 J/(mol·K)) and T is the temperature in Kelvin.
In this case, the given temperature is 366.6 Kelvin. We can substitute the values into the formula:
Kinetic energy per mole = (3/2) * (8.314 J/(mol·K)) * 366.6 K
Now, we can calculate the result:
Kinetic energy per mole = (3/2) * 8.314 J/(mol·K) * 366.6 K
= 36.8766 J/(mol·K) * 366.6 K
= 13,504.5996 J/mol
To convert this result to kJ/mol, we divide by 1000:
13,504.5996 J/mol / 1000 = 13.5046 kJ/mol
Therefore, the kinetic energy per mole of gaseous NH3 molecules at 366.6 Kelvin is approximately 13.5046 kJ/mol.
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Determine the energy released by the fission of U-235 (U-235 becomes Cs-138 and Sr-94, plus neutrons and energy).
Given the B.E./A is as follows:
U-2357.6 MeV
Cs-1388.36 MeV
Sr-948.59 MeV
The energy release by the fission of U-235 is 7.05 × 10⁻¹² J.
The energy released by the fission of U-235 (U-235 becomes Cs-138 and Sr-94, plus neutrons and energy) can be determined by using the Einstein's mass-energy equivalence relation which is given as,
E = (Δm)c²
Here, E is the energy released during the fission of U-235, Δm is the mass defect and c is the speed of light in vacuum. The mass defect can be calculated by subtracting the mass of the nucleus from the sum of the masses of its constituents (protons and neutrons).
The mass of U-235 can be obtained from the atomic mass table which is equal to 235.043923 u.
The mass of Cs-138 is equal to 137.905991 u and the mass of Sr-94 is equal to 93.915360 u.
The mass defect is given by:
Δm = [(mass of reactants) - (mass of products)]×(1.66054 × 10⁻²⁷ kg/u)c²
We get the mass defect to be 0.202064 u.
The energy released is then given by:
E = (Δm)c²E = (0.202064 u)×(1.66054 × 10⁻²⁷ kg/u)×(2.99792 × 10⁸ m/s)²
E = 1.801 × 10⁻¹¹ J/u
To find the total energy released, we need to multiply the energy per unit mass by the mass of U-235 involved in the fission reaction. The mass of U-235 involved in the fission reaction can be calculated as:
mass of U-235 = (number of U-235 nuclei)×(mass of U-235 nucleus)/Avogadro's number
mass of U-235 = (1 mole U-235/Avogadro's number)×(mass of U-235 nucleus)
mass of U-235 = (0.001 kg/6.022 × 10²³)×(235.043923 u)×(1.66054 × 10⁻²⁷ kg/u)
mass of U-235 = 3.912 × 10⁻²⁵ kg
Energy released by the fission of U-235 = (Energy released per unit mass)×(mass of U-235 involved in the fission reaction)
Energy released by the fission of U-235 = (1.801 × 10⁻¹¹ J/u)×(3.912 × 10⁻²⁵ kg)
Energy released by the fission of U-235 = 7.05 × 10⁻¹² J
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