Daniel incorrectly applied the square root to both terms on the left side of the equation and he forgot to take into account the possibility of multiple roots when solving for the cube root of z³.
Let's first look at the first equation that Daniel solved incorrectly:
25x²-16=9
Daniel's solution:
√(25x² )-√16=√9
5x-4=±3
5x=±7
x=±7/5
What Daniel did wrong here is that he incorrectly applied the square root to both terms on the left side of the equation.
Instead, he should have simplified the left side first, using the fact that √(a²) = |a|, before applying the square root. So the correct solution is:
25x²-16=9
25x²=25
x²=1
x=±1
Now let's look at the second equation that Daniel solved incorrectly:
z³-2=6
Daniel's solution:
z³=8
∛(z^3 )=∛8
z=±2
What Daniel did wrong here is that he forgot to take into account the possibility of multiple roots when solving for the cube root of z³.
In fact, z³ = 8 has three roots: z = 2, z = -1 + i√3, and z = -1 - i√3. So the correct solution is:
z³-2=6
z³=8
z=2
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6 yards 2 feet 5 inches equals
Answer:
245 inches
Step-by-step explanation:
Yards->Inches
1 yard = 36 inches
Therefore, 6 yards = 36 x 6 = 216 inches
Feet->Inches
1 feet = 12 inches
Therefore, 2 feet = 2 x 12 = 24 inches
Adding all inches,
=> 216 + 24 + 5
=> 245 inches
what is 6+9f+5+4+3–4f simplified
Answer:
18+5f
Step-by-step explanation:
6+9f+5+4+3-4f
=11+5f+7
=18+5f
Answer:
18 + 5f
Step-by-step explanation:
To simplify the expression 6 + 9f + 5 + 4 + 3 - 4f, we must combine like-terms.
6 + 5 + 4 + 3 = 18 (6,5,4, and 3 are like-terms because they are all normal integers)
18 + 9f - 4f (9f and 4f are like-terms because they have the same variable f)
18 + 5f
A small country emits 80,000 kilotons of carbon dioxide per year. In a recent global agreement, the country agreed to cut its carbon emissions by 2.5% per year for the next 7 years. In the first year of the agreement, the country will keep its emissions at 80,000 kilotons and the emissions will decrease 2.5% in each successive year. How many total kilotons of carbon dioxide would the country emit over the course of the 7 year period, to the nearest whole number?
Using an exponential function and integration, it is found that over the entire 11 year period, the country will emit 763,968 kilotons of carbon dioxide.
Exponential function:
A decaying exponential function is modeled by:
⇒ A (t) = A (0) (1 - r)ⁿ
In which:
A(0) is the initial value.
r is the decay rate, as a decimal.
In this problem:
In the first year, the country emits 80,000 kilotons, hence . A (0) = 8000
Emissions will decrease 2.5% in each successive year, hence r = 0.026
Then, the equation for the amount emitted each year is:
⇒ A (t) = A (0) (1 - r)ⁿ
⇒ A (t) = 80000 (1 - 0.025)ⁿ
⇒ A (t) = 80,000 (0.975)ⁿ
The total amount emitted over the course of the 11 year period is:
T = ∫ 0 to 11 ( A (t) ) dt
Hence:
T = ∫ 0 to 11 ( 80000 (0.974)ⁿ ) dt
Applying the Fundamental Theorem of Calculus:
T = 763,968 kilotons
Hence, Over the entire 11 year period, the country will emit 763,968 kilotons of carbon dioxide.
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Helppppppp anyoneeee?
Answer:
( -6, 6)
Step-by-step explanation:
LESSON 26 SESSION 3
There are at least 12 people on a bus.
a. Write and graph an inequality to show the number of people who may be on
the bus.
An inequality to show the number of people is p ≥ 12
Writing and graphing an inequality to show the number of peopleLet p be the number of people on the bus. We know that there are at least 12 people on the bus, so we can write:
p ≥ 12
This inequality means that the number of people on the bus, p, must be greater than or equal to 12.
To graph this inequality, we can draw a horizontal line at y = 12 on the y-axis, and shade the area above the line, since any value of p that is greater than or equal to 12 satisfies the inequality.
Here's a graph of the inequality attached
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50 Points! Solve the equation or inequality. Please show as much work as possible. Photo attached. Thank you!
The equation 5ʷ ⁺ ³ = 17 when solved for w is approximately w = 0.41
Solving the equations or inequalities for wFrom the question, we have the following parameters that can be used in our computation:
5ʷ ⁺ ³ = 17
Take the logarithm of both sides of the equation
So, we have
w + 3 = ln(17)/ln(3)
Evaluate the quotient on the left side of the equation
This gives
w + 3 = 2.59
So, we have
-3 + w + 3 = 2.59 - 3
Evaluate
w = 0.41
Hence, the equation when solved for w is approximately w = 0.41
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Consider a tech company that wants to offer free breakfast to its employees if their confidence interval shows it will decrease the
proportion of employees who skip breakfast.
Each interval shows the difference in proportion of p₁ - P2 where p₁ represents the employees who skip breakfast when free
breakfast is offered and p2 represents the employees who skip breakfast when free breakfast is not offered. Determine if there is
enough evidence to suggest that offering free breakfast results in an increase in the proportion of employees who don't skip
breakfast.
(-0.44,-0.16)
Yes, we can be confident that employees will skip breakfast less when free breakfast is offered than when it's not.
Or
No, our confidence interval shows that there could be no difference in the proportion of employees who skip breakfast.
(-0.25, 0.05)
Yes, we can be confident that employees will skip breakfast less when free breakfast is offered than when it's not.
Or
No, our confidence interval shows that there could be no difference in the proportion of employees who skip breakfast.
(-0.23, 0.15)
Yes, we can be confident that employees will skip breakfast less when free breakfast is offered than when it's not.
Or
No, our confidence interval shows that there could be no difference in the proportion of employees who skip breakfast.
Answer:
Step-by-step explanation:
In this scenario, the confidence interval represents the possible range of differences in the proportion of employees who skip breakfast when free breakfast is offered and when it's not. A confidence interval that does not include zero indicates that there is statistically significant evidence to suggest a difference in proportions between these two groups.
Therefore, for the first interval (-0.44,-0.16), since it does not include zero, we can be confident that offering free breakfast results in a decrease in the proportion of employees who skip breakfast.
For the second interval (-0.25, 0.05), since it contains zero, we cannot be confident that there is a difference in the proportion of employees who skip breakfast between the two groups.
Similarly, for the third interval (-0.23, 0.15), since it contains zero, we cannot be confident that there is a difference in the proportion of employees who skip breakfast between the two groups.
So, the correct answer is:
For the interval (-0.44,-0.16), we can be confident that employees will skip breakfast less when free breakfast is offered than when it's not.
For the intervals (-0.25, 0.05) and (-0.23, 0.15), our confidence interval shows that there could be no difference in the proportion of employees who skip breakfast.
7: An entrance examination for a job consists of 25% English, 30% Mathematics, 25% computer and 20% accounting, and the passing mark is 45. If an applicant sat for the examination and scored 60% in English, 35% in Mathematics, 80% in computer and 50% in Accounting, do you think she will be accepted?
Answer: The overall score obtained is 55.5, which is higher than the passing mark of 45. Therefore, the applicant will be accepted.
Step-by-step explanation:To determine if the applicant will be accepted, we need to calculate their overall score based on the percentage weights and passing mark given for each subject.
The overall score can be calculated as follows:
Overall score = (0.25 x English score) + (0.3 x Math score) + (0.25 x Computer score) + (0.2 x Accounting score)
Substituting the given scores, we get:
Overall score = (0.25 x 60) + (0.3 x 35) + (0.25 x 80) + (0.2 x 50)
= 15 + 10.5 + 20 + 10
= 55.5
example of a multi-step combination problem
Answer:
A multi-step-word problem is like a puzzle with lots of pieces. Multi-step word problems are math problems that have more than one operation. An operation is addition, subtraction, multiplication, or division. Multi-step word problems can have any combination of these operations
The test scores of 40 students are listed below.
30 35 43 44 47 48 54 55 56 57
59 62 63 65 66 68 69 69 71 72
72 73 74 76 77 77 78 79 80 81
81 82 83 85 89 92 93 94 97 98
a. Find the standard deviation and the variance for the data
b find the five-number summery for the data
c construct a boxplot for the given data. Include the value of the 5-number summery in the boxplot
a. the standard deviation is approximately 25.01 and the variance is approximately 625.21.
b.the five-number summary for the data is: 30, 57, 71, 81, 98.
what is standard deviation?The standard deviation is a measurement of how widely apart a group of data points are from the mean (average) value. You can see how far each data point deviates from the mean value. When the standard deviation is minimal, the data points are densely packed around the mean; when it is great, the data points are more widely spaced from the mean. The standard deviation is a statistical concept that is frequently used to compare the degree of variability across various data sets as well as to infer and make conclusions about a population from a sample.
a. To find the standard deviation and variance for the given data, we can use the following formulas:
Variance: σ² = ∑(x - μ)² / n
Standard deviation: σ = sqrt(σ²)
where x is the data point, μ is the mean of the data, and n is the sample size.
First, we need to find the mean of the data:
μ = (30 + 35 + 43 + 44 + 47 + 48 + 54 + 55 + 56 + 57 + 59 + 62 + 63 + 65 + 66 + 68 + 69 + 69 + 71 + 72 + 72 + 73 + 74 + 76 + 77 + 77 + 78 + 79 + 80 + 81 + 81 + 82 + 83 + 85 + 89 + 92 + 93 + 94 + 97 + 98) / 40
μ = 69.3
Now, we can use the variance formula:
σ² = ∑(x - μ)² / n
σ² = [(30 - 69.3)² + (35 - 69.3)² + ... + (98 - 69.3)²] / 40
σ² = 625.21
Finally, we can find the standard deviation:
σ = sqrt(σ²)
σ = sqrt(625.21)
σ ≈ 25.01
Therefore, the standard deviation is approximately 25.01 and the variance is approximately 625.21.
b. To find the five-number summary, we need to find the minimum, maximum, median, and quartiles (Q1 and Q3) of the data:
Minimum: 30
Q1: 57
Median: 71
Q3: 81
Maximum: 98
Therefore, the five-number summary for the data is: 30, 57, 71, 81, 98.
c. To construct a boxplot for the given data, we can use the five-number summary:
Draw a number line and mark the minimum, Q1, median, Q3, and maximum.
Draw a box from Q1 to Q3.
Draw a vertical line inside the box at the median.
Draw whiskers from the ends of the box to the minimum and maximum, or to the nearest data point within 1.5 times the interquartile range (IQR).
Mark any outliers outside the whiskers.
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Matthew is working two summer jobs, making $9 per hour babysitting and making $13 per hour landscaping. In a given week, he can work a maximum of 16 total hours and must earn at least $180. If Matthew worked 3 hours babysitting, determine the maximum number of whole hours landscaping that he can work and still meet his requirements. If there are no possible solutions, submit an empty answer.
Answer:
Step-by-step explanation:
3 hours of babysitting = $27
if he needs $180 per week he needs to work at least 12 hours in landscaping.
12 hours of Landscaping = $156
156 + 27 = 183
It cuts down about an hour of work.
If I misunderstood something, please let me know. Have a good day :)
Nevaeh works in a department store selling clothing. She makes a guaranteed salary of $300 per week, but is paid a commission on top of her base salary equal to 20% of her total sales for the week. How much would Nevaeh make in a week in which she made $1975 in sales? How much would Nevaeh make in a week if she made x dollars in sales?
a) When Nevaeh makes $1,975 in sales for the week, her total earnings, based on her base salary and sales commission rate, are $695.
b) When she makes x in sales for the week, her total earnings are represented by the equation, y = 300 + 0.2x.
How the total earnings are computed:The total earnings for the week can be determined using the equation y = 300 + 0.2x, where x represents her total sales for the week while 0.2 represents a 20% sales commission.
The base salary, in this situation, is $300.
Guaranteed (base) salary per week = $300
Sales commission = 20% of sales
Sales for the week = $1,975
a) Total earnings = $695 ($300 + $1,975 x 20%)
Sales for the week = x
b) Total earnings, y = $300 + 0.2x ($300 + 0.2x)
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Can someone help me please
Chapter 13/14. The Civil War. Watch the Hari Jones TED talk. Then, answer the following questions in 7-10 sentences. Then, respond to two different classmates' posts by asking a question or challenging their assertions (respectfully). Try to make sure everyone has a response to their post. Finally, respond to the question or assertion your classmate(s) replied to you. Responses are meant to acknowledge what the other student has communicated, and either expand on their ideas, or push back on them, then offer positive feedback. Consider using the following template: "Hi _____. I was interested in ______ about your post because _______. Also/However, ______. I think ______ because _____. Do you think/feel _____/? I feel _____. "
In serving in the military, what contributions did African Americans make to the Union war effort?
What kinds of discrimination did they face? What kind of heroism did they exhibit, and when?
Write a reaction to Dr/ Jones' TED talk. What did you know/not know? What surprised/amazed/angered/made you proud/saddened/made you happy?
Hi Annie. I was interested in engaging you about your post because of my differing view. However, I think I am open to changing my views because I am not dogmatic. Do you think I am too pliable?
How did African Americans contribute?Throughout the Civil War, African Americans assumed crucial roles in the Union war effort. From enlisting as soldiers to acting as laborers and even spies, their contributions left an indelible impact on history.
Despite being subjected to rampant discrimination and racism from both Confederacy and Union forces, these individuals exhibited incredible heroism and bravery.
Excelling in dangerous missions that often led them to become prisoners of war, they persevered as they fought for the freedoms of themselves and others.
Hari Jones' TED talk shines a much-needed light on this overlooked aspect of history, revealing not only the extensive involvement of African American soldiers but also the pivotal role played by women in areas such as nursing, cooking, and espionage.
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A scale drawing for an apartment is shown below. In the drawing, 3cm represents 4cm. Assuming the bedroom is rectangular, find the area of the real room
The calculated value of the area of the real room is 10.67 cm²
Finding the area of the real roomFrom the question, we have the following parameters that can be used in our computation:
In the drawing, 3cm represents 4cm
This means that
Scale factor = 4/3
Also, we have the dimension of the scale drawing to be
Dimension = 3cm by 2 cm
So, the area of the bedroom in the scale is
Area = 3cm * 2cm
For the actual room, we have
Area = 3cm * 2cm * (4/3)²
Evaluate the exponent
Area = 3cm * 2cm * (16/9)
Evaluate the products
Area = 10.67 cm²
This means that the real area is 10.67 cm²
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Ignatio and his two friends each bought a ticket to play lazer tag and each spent $15 on game tokens at Fun Mountain. Let t represent the cost of a lazer tag ticket. Enter an expression that represents the total amount that Ignatio and his friends spent.
The expression that represents the total amount that Ignatio and his friends spent is 15t
From the question, we have the following parameters that can be used in our computation:
Each spent $15 on game tokens at Fun MountainEach cost = tAfter spending on t games, we have
f(t) = Rate * t
Substitute the known values in the above equation, so, we have the following representation
f(t) = 15 * t
Evaluate
f(t) = 15t
Hence, the expression that represents the total amount that Ignatio and his friends spent is 15t
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17. The Amethyst Woodstar species of
hummingbird beats its wings about
80 times per second. If there are
60 seconds in a minute, about how
many times does the Amethyst
Woodstar beat its wings in
9 minutes?
A 12,600
B 37,800
C50,400
D 43,200
If the Amethyst Woodstar species of hummingbird beats its wings about 80 times per second and there are 60 seconds in a minute, it beats its wings D. 43,200 times in 9 minutes.
How is the number of times determined?The number of times that the Amethyst Woodstar species of hummingbird beats its wings in 9 minutes can be determined using the mathematical operation of multiplication.
The number of times it beats its wings = 80 seconds per second
The total number of minutes involved = 9 minutes
1 minute = 60 seconds
The total number of times it beats in 9 minutes = 43,200 (80 x 60 x 9)
Thus, using multiplication operation, the number of times the bird beats its wings is Option D.
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pls pls help me with answer(correct it)
Suppose a stock drops in value by 60% one week then increases in value the next week by 75% is the value higher or lower then when u started and by how much
The value of the stock at the end of the two weeks would be 70, which is 30% lower than the starting value of 100.
What is Algebraic expression ?
An algebraic expression is a combination of variables, constants, and mathematical operations such as addition, subtraction, multiplication, division, exponents, and roots. It typically represents a mathematical relationship or formula and can be used to solve problems and make predictions.
Assuming you started with a value of 100, the value of the stock would be lower after these two weeks.
In the first week, the stock dropped by 60%, which means its value decreased to 40% of its original value. Therefore, the value of the stock at the end of the first week would be:
100 * 0.4 = 40
In the second week, the stock increased in value by 75%. Therefore, the value of the stock at the end of the second week would be:
40 * 1.75 = 70
Therefore, the value of the stock at the end of the two weeks would be 70, which is 30% lower than the starting value of 100.
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solve the rational equation 2/x-2+3/x-4=1/x^2=6x+8
the solution to the original rational equation is: [tex]x = 4.678[/tex] (rounded to three decimal places)
What is the rational equation?A rational number is a number that can be expressed as a ratio of two integers, where the denominator is not zero. In other words, it is a number that can be written in the form of a/b, where a and b are integers and b is not equal to zero
According to InformationThere are two equal signs in the equation, which is incorrect. Assuming you meant to write:
[tex]2/(x-2) + 3/(x-4) = 1/(x^2 + 6x + 8)[/tex]
We can start by finding a common denominator for the left-hand side of the equation:
[tex]2/(x-2) + 3/(x-4) = 1/[(x+2)(x+4)][/tex]
Multiplying both sides of the equation by (x-2)(x-4)(x+2)(x+4), we get:
[tex]2(x-4)(x+2)(x+4) + 3(x-2)(x+2)(x+4) = (x-2)(x-4)[/tex]
Expanding and simplifying, we get:
[tex]5x^3 - 19x^2 - 39x + 56 = 0[/tex]
This polynomial equation does not factor nicely, so we can use the rational root theorem or numerical methods to find approximate solutions. Using a calculator or computer, we find that there is one real solution to the equation:
[tex]x \approx 4.678[/tex]
Therefore, the solution to the original rational equation is:
[tex]x = 4.678[/tex] (rounded to three decimal places)
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PLEASE HELP SO CONFUSING PLEASE HELP WITH THE WHOLE PROBLEM THOUGH‼️‼️
When a student is chosen at random, the student is more likely to be from Mrs Johnson's class.
The probability that the student got A in the first semester will be 0.15.
How to calculate the probabilityWhen a student is chosen at random, the student is more likely to be from Mrs Johnson's class. It should be noted that this is because 152 students out of 300 are from her class compared to Dixon who has 148 students.
The probability that the student got A in the first semester will be:
= 46 / 300
= 0.15
The probability for the last question will be:
= 14 / 148
= 0.095.
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Find the measure of the angle between the two vectors. u = -7i and v = -4i + 4j
thickness of piece of metal in m. 0.00053
The thickness of the said metal piece is 560 micrometers
How to solveSolving this question would require us to switch the thickness of the metal from meters to micrometers.
So, 1 meter is equivalent to 1,000,000 micrometers; and when 0.00053 meters is multiplied by 1,000,000, we get:
0.00053 m * 1,000,000 = 560 µm
Consequently, the thickness of the said metal piece is 560 micrometers.
A metal piece's thickness is defined as the distance, measured perpendicular to its opposing surfaces, between those surfaces. It is a crucial aspect in determining the strength and longevity of the metal object and is often stated in length measures like millimeters, inches, or micrometers.
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What is the thickness of the metal piece in micrometers, given that it is 0.00053 meters thick?
Pre calculus trigonometry
The result of the division of the two complex numbers is equal to z₁ / z₂ = (8 / 9) · [cos (π / 10 - π / 12) + i sin (π / 10 - π / 12)].
How to find the division of two complex numbers
In this problem we find two complex numbers in rectangular form, whose division must be found. This can done by easily by means of complex numbers in polar form and definition of division:
Complex number (rectangular form)
z = r · (cos θ + i sin θ)
Complex number (polar form)
[tex]z = r\cdot e^{i\cdot \theta}[/tex]
Where:
r - Normθ - DirectionDivision between two complex numbers in polar form:
[tex]\frac {z_{1}}{z_{2}} = \left(\frac{r_{1}}{r_{2}}\right)\cdot e^{i\cdot (\theta_{1}-\theta_{2})}[/tex]
This number is equivalent to the following expression in rectangular form:
z₁ / z₂ = (r₁ / r₂) · [cos (θ₁ - θ₂) + i sin (θ₁ - θ₂)]
If we know that r₁ = 8, r₂ = 9, θ₁ = π / 10 and θ₂ = π / 12, then the division of the two complex numbers is:
z₁ / z₂ = (8 / 9) · [cos (π / 10 - π / 12) + i sin (π / 10 - π / 12)]
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Which transformation would move Point A(-5, -2) to Quadrant IV? Mark all that apply.
To move Point A(-5, -2) to Quadrant IV, we need to apply one or more transformations that result in a reflection about the x-axis and/or a rotation of 180 degrees. The following transformations would move Point A to Quadrant IV:
Reflection about the x-axis: This transformation would change the sign of the y-coordinate, making it positive. So, the image of A after this transformation would be (−5, 2).
Rotation of 180 degrees about the origin: This transformation would change the sign of both the x and y-coordinates, making them positive. So, the image of A after this transformation would be (5, 2).
Therefore, either a reflection about the x-axis or a rotation of 180 degrees would move Point A to Quadrant IV.
It’s similar figure with proportions
Answer:
x = 72 cm
Step-by-step explanation:
Because the two figures are similar, we can create a proportion between one set of similar sides (i.e., the side measuring 48 cm and the side measuring 6 cm) and another set (the side measuring x cm and the side measuring 9 cm. This, will allow us to solve for x:
[tex]\frac{6}{48}=\frac{9}{x}\\ 432=6x\\ 72=x[/tex]
Another way to solve would have been to notice that the scale factor used to create the bigger puzzle piece is 8 since 6 * 48. In order for the smaller figure to be considered similar to the larger, every side had to be multiplied by the same scale factor. Since the scale factor is 6, x must be 72 cm as 9 * 8 = 72
The combined math and verbal scores for females taking the SAT-I test are normally distributed with a mean of 998 and a standard deviation of 202 (based on date from the College Board). If a college includes a minimum score of 1025 among its requirements, what percentage of females do not satisfy that requirement?
I have already tried 64.8% as my answer!!
The percentage of females that do not satisfy the requirement is given as follows:
55.17%.
How to obtain probabilities using the normal distribution?The z-score of a measure X of a normally distributed variable that has mean represented by [tex]\mu[/tex] and standard deviation represented by [tex]\sigma[/tex] is obtained by the equation presented as follows:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The z-score represents how many standard deviations the measure X is above or below the mean of the distribution of the data-set, depending if the obtained z-score is positive(above the mean) or negative(below the mean).The z-score table is used to obtain the p-value of the z-score, and it represents the percentile of the measure X in the distribution.The mean and the standard deviation are given as follows:
[tex]\mu = 998, \sigma = 202[/tex]
The proportion that does not satisfy the requirement is the p-value of Z when X = 1025, hence:
Z = (1025 - 998)/202
Z = 0.13
Z = 0.13 has a p-value of 0.5517.
Hence the percentage is given as follows:
0.5517 x 100% = 55.17%.
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please hurry thats all i ask
The value of A, B, C,D are 33, 36,25 and 16 respectively
What is word problem?A word problem is a math problem written out as a short story or scenario. We use this word problem to solve the unknowns.
The total number of students is 60 and 24 students prefer volley ball. therefore the number of students that prefer basketball = 60-24 = 36 students
Since 11 student are girls that prefer basketball, therefore the number of boys that prefers basketball = 36-11 = 25
C = 25
and B = 36
55% of the students are boys, therefore the number of boys = 55/100 × 60 = 33
A = 33
therefore the number of boys that prefer volley ball= 33-25 = 8
Number of girls = 60-33 = 27
therefore number of girls that prefer volley ball = 27-11 = 16
D= 16
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2 1/2 - 4 1/4
how do I slove this
The fraction expression 2 1/2 - 4 1/4 when evaluated is -1 3/4
Evaluating the fraction expressionTo solve 2 1/2 - 4 1/4, we need to convert the mixed numbers into improper fractions, then find a common denominator and subtract the two fractions.
First, we convert 2 1/2 and 4 1/4 into improper fractions:
2 1/2 = 5/2
4 1/4 = 17/4
Next, we find a common denominator. In this case, the smallest common denominator is 4:
5/2 = 10/4
17/4 = 17/4
Now we can subtract the fractions:
10/4 - 17/4 = -7/4
Finally, we can convert the resulting improper fraction back into a mixed number:
-7/4 = -1 3/4
Therefore, 2 1/2 - 4 1/4 = -1 3/4.
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The hypotenuse of an isosceles right triangle is 13 centimeters longer than either of its legs. Find the exact length of
each side. (Hint: An isosceles right triangle is a right triangle whose legs are the same length.)
The length of one leg is
the length of the other leg is
(Simplify your answers. Type exact answers, using radicals as needed.)
and the length of the hypotenuse is
The length of the other legs of the isosceles right triangle is 13 + 13√2 centimetres.
How to find the sides of a right triangle?The hypotenuse of an isosceles right triangle is 13 centimetres longer than either of its legs.
An isosceles right triangle is a right triangle whose legs are the same length. The sides of a right triangle can be found using Pythagoras's theorem.
Therefore,
let
y = other legs
Therefore,
(x + 13)² = x² + x²
(x + 13)(x + 13) = 2x²
x² + 13x + 13x + 169 = 2x²
x² - 26x - 169 = 0
Therefore, using the quadratic formula,
[tex]\frac{-b+\sqrt{b^{2} - 4ac } }{2a}[/tex]
where
a = 1b = -26c = -169x = 13 ± 13√2
Therefore, we can only use positive values
x = 13 + 13√2
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Consider the following graph: What is the average slope/rate of change between (0, 1) and (2, 4)? 2/3 What is the average slope/rate of change between (-2, 1/4) and (-1, 1/2)? -1/4 Is the slope/rate of change constant (not changing/the same)? No Is the function linear? No
a) The average slope or rate of change between (0, 1) and (2, 4) is 3/2.
b) The average slope or rate of change between (-2, 1/4) and (-1, 1/2) is 1/4.
c) The slope or rate of change is not constant between these two pairs of points, since the average slopes are different.
d) The function connecting these pairs of points is not a linear function.
Define a slope?A line's steepness and direction are measured by the line's slope. Without actually using a compass, determining the slope of lines in a coordinate plane can assist in forecasting whether the lines are parallel, perpendicular, or none at all.
Any two different points on a line can be used to calculate the slope of any line. The ratio of "vertical change" to "horizontal change" between two different locations on a line is calculated using the slope of a line formula.
The average slope or rate of change between two points (x1, y1) and (x2, y2) on a line is given by:
average slope = (y2 - y1) / (x2 - x1)
For the points (0, 1) and (2, 4), the average slope is:
average slope = (4 - 1) / (2 - 0) = 3/2
For the points (-2, 1/4) and (-1, 1/2), the average slope is:
average slope = (1/2 - 1/4) / (-1 - (-2)) = 1/4
When the average slopes of these two pairs of points differ, the slope or rate of change is not constant. As a result, the function connecting these point pairs is not linear.
Because a linear function has a constant slope, a function with a variable slope cannot be linear.
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