Identify the differentiation rule needed in order to find the derivative of
each of the following functions with respect to x.
a) y = sinxcosx
b) y = sin(cosx)
c) y = sin-x
d) y = sinx + cosx

Answer:

55 degrees

Step-by-step explanation:

sin Φ = cos 35

sin Φ = .81915

Φ = arc sin (.81915) = 55 degrees

Or you could just know that the cos of an angle is equal to the sin of its comeplement

The derivative of the function f(x) = 2 \sqrt(x) / (8x² + 3x - 9) evaluated at x = 4.

To find f'(x), we need to differentiate the given function f(x) using the power rule and the chain rule of differentiation.

First, we can rewrite the function f(x) as:

f(x) = 2x^{1/2} / (8x² + 3x - 9)

Next, we can differentiate f(x) with respect to x:

f'(x) = d/dx [2x^{1/2} / (8x² + 3x - 9)]

Using the quotient rule of differentiation, we have:

f'(x) = [ (8x² + 3x - 9) d/dx [2x^{1/2}] - 2x^{1/2} d/dx [8x² + 3x - 9] ] / (8x² + 3x - 9)²

Applying the power rule of differentiation, we have:

f'(x) = [ (8x² + 3x - 9)(1/2) - 2x{1/2}(16x + 3) ] / (8x² + 3x - 9)²

Now we can evaluate f'(x) at x = 4 by substituting x = 4 into the expression for f'(x):

f'(4) = [ (8(4)² + 3(4) - 9)(1/2) - 2(4)^(1/2)(16(4) + 3) ] / (8(4)² + 3(4) - 9)²

f'(4) = [ (128 + 12 - 9)(1/2) - 2(4)^(1/2)(67) ] / (128 + 12 - 9)^2

f'(4) = [ 131^(1/2) - 2(4)^(1/2)(67) ] / 12167

Therefore, f'(4) = [ 131^(1/2) - 134(2)^(1/2) ] / 12167.

This is the value of the derivative of f(x) at x = 4.

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The equation for an exponential that models this data set is: p = 10(2)^t

To determine an equation for an exponential that models the given data set in the form p = a(b)^t, we first need to identify the values of a and b. To do this, we can use two points from the data set and solve for a and b. Let's choose the points (0, 10) and (2, 40):

When t = 0, p = 10: 10 = a(b)^0 = a
When t = 2, p = 40: 40 = a(b)²

Dividing the second equation by the first, we get:

4 = (b)²

Taking the square root of both sides, we get:

b = 2

Now that we have the value of b, we can use one of the original equations to solve for a:

10 = a(2)^0 = a

So, a = 10.

Therefore, the equation for an exponential that models this data set is:

p = 10(2)^t

We can check this equation by plugging in the other data points and verifying that they satisfy the equation. And rounding b to the nearest hundredth gives us b = 2.00.

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Thus, the height of cone for the given values of circumference an f volume is found as: 4 cm.

A tri shape that resembles a cone is what is known as a conical shape. A cone has a flat end that gradually taper towards a single point at the top known as the apex. Most commonly, a conical shape's flat end has an oval or circular shape. Conical shapes are on your mind when you imagine an ice cream cone with only a pointed end.

Volume of a cone = 1/3 * π *r²*h

Given that:

using circumference c = 6π

c = 2πr (for circular base)

6π  = 2πr

r = 3 cm

Now, using the volume;

Volume of a cone = 1/3 * π *r²*h

1/3 * π *3²*h = 12π

3h = 12

h = 4 cm

Thus, the height of the cone for the given values of circumference an f volume is found as: 4 cm.

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Jones would save $20 over 2 months by choosing Texas Bank over Lone Star Bank for his online banking needs.

To answer this question, we need to know the fees and charges associated with online banking at Texas Bank and Lone Star Bank. Without this information, it's impossible to accurately calculate how much Jones would save by choosing one bank over the other.

Assuming we have this information, we can use the following steps to calculate Jones' potential savings:

Calculate the fees and charges associated with 40 transactions per month at each bank.

Subtract the total fees and charges at Texas Bank from the total fees and charges at Lone Star Bank to determine the difference.

Multiply the difference by 2 to determine how much Jones would save in 2 months by choosing Texas Bank over Lone Star Bank.

For example, let's say that the fees and charges at Lone Star Bank for 40 transactions per month total $20, while the fees and charges at Texas Bank for the same number of transactions total $10. In this case, the difference would be $10 per month.

To calculate Jones' potential savings over 2 months, we would multiply $10 by 2, giving us a total potential savings of $20.

So, in this hypothetical scenario, Jones would save $20 over 2 months by choosing Texas Bank over Lone Star Bank for his online banking needs.

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To synthesize information regarding when books were written, Alex should create a timeline.  So, the correct option is A.

A timeline is a visual representation of chronological events, which can be used to arrange information in order based on their time of occurrence. In this case, Alex can plot the publication dates of the books on a timeline, allowing him to see how the dates relate to each other and to other events. This will help him to identify patterns and trends in the publication history of the books.

A Venn diagram, on the other hand, is a tool used to compare and contrast two or more sets of information. It is not well-suited for presenting chronological information.

A chart may be useful in presenting data in a visual manner, but it may not be as effective as a timeline in showing the order of events over time.
Therefore, the best answer from the choices provided is A. timeline.

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The volume of the composite figure is 96.3 cm³

First we need to find the volume of the cylinder, and then remove the volume of the rectangular prism.

The radius of the prism is 3cm and the height is 5cm, then the volume is:

V = 3.14*R²*H

V = 3.14*(3cm)²*5cm

V = 141.3 cm³

And the volume of the prism is:

V' = 3cm*3cm*5cm = 45 cm³

The difference gives:

volume = 141.3 cm³ -  45 cm³

volume = 96.3 cm³

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On the basic of square peg sydney smith written point the probability that it is im-possible to fit a square peg in a round hole is equals to the zero.

Probability is defined as the number of chances of occurrence of an event. It is the ratio of the number of favorable outcomes to the total outcomes in that sample space. Mathematical formula is written as, probability of an event, P(E) = (Number of favorable outcomes)/(Total possible outcomes). Now, we have specify that according to sydney smith written in on the conduct of the understanding about square peg that it is impossible to fit a square peg in a round hole. Let consider an event A of fit a square peg in a round hole. We have specify that it is impossible to fit a square peg in a round hole. So, the favourable possible outcomes for event A = 0. Therefore, the probability that to fit a square peg in a round hole, P(A) = 0

Hence, required probability value is zero.

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Leslie’s annual social security deduction is $3551.70.

Word problems are sentences describing a 'real-life' situation where a problem needs to be solved by way of a mathematical calculation.

Leslie’s  annual salary is $57,285. 50 and her annual social security deduction is 6.2% of the annual salary. We can say:

Annual social security deduction = 6.2/100 * 57,285. 50

Annual social security deduction = $3551.70

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The integral of ſ(cosa + 3x^4) dx simplifies to ∫cos(x) dx, which can be evaluated as sin(x) + C, where C is the constant of integration.To evaluate the integral of ſ(cosa + 3x^4) dx using even and odd functions, we can decompose the integrand into even and odd parts.

Let's first identify the even and odd parts of the integrand. The function cos(x) is an even function because it is symmetric with respect to the y-axis, i.e., cos(-x) = cos(x). On the other hand, the function 3x^4 is an odd function because it is symmetric with respect to the origin, i.e., (-x)^4 = x^4.

We can rewrite the integrand as a sum of even and odd functions:

cos(x) + 3x^4 = (1/2) * (cos(x) + cos(-x)) + (1/2) * (3x^4 - 3(-x)^4)

Now, we can use the properties of even and odd functions to simplify the integral. The integral of an even function over a symmetric interval is equal to twice the integral of the function over half of the interval. Similarly, the integral of an odd function over a symmetric interval is equal to zero.

So, the integral of (1/2) * (cos(x) + cos(-x)) dx is equal to (1/2) * 2 * ∫cos(x) dx, since cos(x) is an even function.

And the integral of (1/2) * (3x^4 - 3(-x)^4) dx is equal to (1/2) * 0, since 3x^4 - 3(-x)^4 is an odd function and the interval of integration is symmetric.

Therefore, the integral of ſ(cosa + 3x^4) dx simplifies to ∫cos(x) dx, which can be evaluated as sin(x) + C, where C is the constant of integration.

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Given that 25% of students solved at least two tasks and there were 32 students who wrote the test, we can prove that there was at least one task that no more than 12 students solved.

There was at least one task that no more than 12 students solved, we can use a proof by contradiction.

Assume that all five tasks were solved by more than 12 students. This means that for each task, there were at least 13 students who solved it. Since there are five tasks in total, this implies that there were at least 5 * 13 = 65 students who solved the tasks.

However, we are given that only 25% of students solved at least two tasks. If we let the number of students who solved at least two tasks be S, then we can write the equation:

S = 0.25 * 32

Simplifying, we find that S = 8.

Now, let's consider the remaining students who did not solve at least two tasks. The maximum number of students who did not solve at least two tasks is 32 - S = 32 - 8 = 24.

If all five tasks were solved by more than 12 students, then the total number of students who solved the tasks would be at least 65. However, the maximum number of students who could have solved the tasks is 8 (those who solved at least two tasks) + 24 (those who did not solve at least two tasks) = 32.

This contradiction shows that our initial assumption is false. Therefore, there must be at least one task that no more than 12 students solved.

Hence, we have proven that there was at least one task that no more than 12 students solved if 32 students wrote the test.

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The correct option is D, The best measure of variability to use to compare the data sets is the standard deviation is the best measure because both data distributions are symmetric.

Let's examine each of the possibilities we've been provided so we can select the best one.

A. Since the data sets have the same minimum weight, the mean is the best indicator.

Option A is untrue about the measure of variability since the mean measures the central tendency of a data collection.

B. The distribution of zucchini weights is tilted to the left, making the range the most accurate measurement.

Given that both of our box plots are symmetric, option B cannot be true, as can be seen.

C. Because the medians of the data sets differ, the median is the best indicator.

Since the median is a measure of a data set's central tendency rather than variability, option C is the correct answer. not true.

D. Because both data distributions are symmetric, the standard deviation is the most accurate measurement.

Option D is the best option since standard deviation is the best measurement for symmetric data sets and both of our provided box plots are symmetric.

Distributions can take on many different forms, depending on the type of random variable being described. Some common distributions include the normal distribution, the uniform distribution, and the binomial distribution. In statistics, a distribution is a function that describes the probabilities of different outcomes in a random variable. A random variable is a variable whose value is determined by chance, such as the outcome of a coin toss or the height of a randomly selected person.

The normal distribution is perhaps the most well-known and is often used to model real-world phenomena, such as heights or weights of people, IQ scores, or measurements of physical characteristics. Understanding distributions is important in statistics because they allow us to make predictions and draw conclusions about populations based on a sample of data.

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Complete Question:-

The box plots show the distributions of weights of cucumbers and zucchini collected from a garden.

(see attachment)

Which statement explains the best measure of variability to use to compare the data sets?

A.The mean is the best measure because the data sets have the same minimum weight.

B. The range is the best measure because the distribution of zucchini weights is skewed left.

C. The median is the best measure because the data sets have different medians.

D. The standard deviation is the best measure because both data distributions are symmetric.

The final cost of the basketball, including sales tax, is $15.84.

The problem states that a new basketball costs $22.00, but it is on sale for 40% off. This means that the customer can purchase the basketball at a discount of 40% from its original price of $22.00.

To calculate the amount of discount, we multiply the original price of the basketball by the discount rate as a decimal:

Discount = 0.40 x $22.00 = $8.80

So, the sale price of the basketball would be:

Sale price = $22.00 - $8.80 = $13.20

Next, the problem states that the sales tax is 5%. Sales tax is a percentage of the sale price of the item, and it is added to the sale price to calculate the final cost of the item.

To calculate the amount of sales tax, we multiply the sale price by the sales tax rate as a decimal:

Sales tax = 0.05 x $13.20 = $0.66

Finally, to calculate the final cost of the basketball, we add the sale price and the sales tax:

Final cost = $13.20 + $0.66 = $15.84

Therefore, the final cost of the basketball, including sales tax, is $15.84.

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the side length of the large square is equal to 4 divided by the square root of π times the side length of the small square.

what is length  ?

Length is a physical quantity that refers to the measurement of a one-dimensional distance or extent, such as the distance between two points. It is typically measured in units such as meters, feet, inches, or centimeters. Length can be used to describe the size or dimensions

In the given question,

Let's assume that the side length of the small square is equal to the diameter of each small bubble.

When four bubbles in squares collide and merge into one large bubble in a square, the total area of the small squares is equal to the area of the large square. Since the side length of each small square is equal to the diameter of the small bubble, the area of each small square is equal to the square of the diameter of the small bubble.

So, if we let d be the diameter of each small bubble, then the area of each small square is equal to d². Therefore, the total area of the four small squares is equal to 4d², and the area of the large square is equal to the sum of the areas of the four small squares, which is 4d².

The area of a circle is equal to πr², where r is the radius of the circle. If we let R be the radius of the large bubble, then its area is equal to π².

We know that the area of the large square is equal to the area of the large bubble, so we have:

4d² = πR²

Solving for R, we get:

R =√(4d²/π)

R = 2d/√(π)

Since the side length of the large square is equal to twice the radius of the large bubble, we have:

Side length of large square = 2R = 4d/√(π)

Therefore, the side length of the large square is equal to 4 divided by the square root of π times the side length of the small square.

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The rate of change of:

total revenue is $500 per day

total cost is $40 per day

total profit is $460 per day

To find the rate of change of total revenue, cost, and profit with respect to time, we need to take the derivative of the given functions with respect to x and then multiply by the rate of change of x with respect to time (dx/dt).

Total revenue (R) = 60x – 0.5x²
dR/dx = 60 – x
When x = 35, dR/dx = 60 – 35 = 25
Rate of change of total revenue = (dR/dx) * (dx/dt) = 25 * 20 = $500 per day

Total cost (C) = 2x+ 10
dC/dx = 2
When x = 35, dC/dx = 2
Rate of change of total cost = (dC/dx) * (dx/dt) = 2 * 20 = $40 per day

Total profit (P) = R - C
dP/dx = (dR/dx) - (dC/dx)
When x = 35, dP/dx = 25 - 2 = 23
Rate of change of total profit = (dP/dx) * (dx/dt) = 23 * 20 = $460 per day

Therefore, the rate of change of total revenue is $500 per day, the rate of change of total cost is $40 per day, and the rate of change of total profit is $460 per day.

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The solution to the equation is x = 15.

To rewrite the exponential equation using the same base, we need to express both 8 and 32 as powers of the same base. Since both 8 and 32 are powers of 2, we can rewrite the equation as:

[tex](2^3)^(2x) = (2^5)^(x+3)[/tex]

Here, we used the fact that[tex](a^b)^c = a^(b*c)[/tex]to simplify the exponents. We also used the property that 8 is equal to 2 raised to the power of 3, and 32 is equal to 2 raised to the power of 5.

Now that we have rewritten the equation with the same base, we can equate the exponents on both sides of the equation to solve for x:

[tex]2^(6x) = 2^(5x + 15)[/tex]

Since the bases on both sides of the equation are equal, we can equate the exponents and solve for x:

6x = 5x + 15

Subtracting 5x from both sides, we get:

x = 15

Therefore, the solution to the equation is x = 15.

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The correct option is the last one, the surface is 3425 m²

Remember that for a sphere of radius R, the volume is:

V = (4/3)pi*R³

S = 4pi*R²

Where pi = 3.14

Here the volume is  6,000π m³, then the radius will be:

R =∛( (3/4)*6,000m³)

R = 16.51 m

Then the surface area is:

[tex]S = 4*3.14*( 16.51 m)^2 = 3,424 m^2[/tex]

The option that is closser to it is the fourth one, so that is the correct option.

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Answer: That made no sense

Step-by-step explanation:

The function table that matches the given rule output = input - 3 is

Input = -2, 1, 6 and output = -5, -2, 3

A) first function table

Output = Input - 3

Value of input:- -2

Putting the value of the input

Output = -2 -3

The value of output we get

Output = -5

Value of input:- 1

Putting the value of the input

Output = 1 -3

The value of output we get

Output = -2

Value of input:- 6

Putting the value of the input

Output = 6 -3

The value of output we get

Output = 3

Hence function table A is correct match

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The function h is given by h(x)=log2(x² -6). The positive value of x that makes h(x) equal to 4 is approximately 4.69

We have the function:

h(x) = log2(x² - 6)

We want to find the value of x that makes h(x) equal to 4:

h(x) = 4

log2(x² - 6) = 4

We can rewrite this equation as:

2⁴ = x² - 6

16 = x² - 6

x²= 22

x = √22 (because we are looking for a positive value of x)

Therefore, the positive value of x that makes h(x) equal to 4 is approximately 4.69 (rounded to two decimal places).

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To simplify the expressions and arrange the terms by their degree, we can write:

$\longrightarrow\sf\textbf\:f(x)\:= -x^4\:+\:2x^2\:-\:5x\:+\:3$

$\longrightarrow\sf\textbf\:=\:-x^4 + 2x^2 - x - 4x + 3$

$\longrightarrow\sf\textbf\:=\:-x(x^3 - 2x + 1) - 4x + 3$

$\longrightarrow\sf\textbf\:g(x)\:=\:x^4 + 4x^2 + 2x + 1$

$\longrightarrow\sf\textbf\:x^4 + 4x^2 + 2x + 1 - 2 + 2$

$\longrightarrow\sf\textbf\:x^4 + 4x^2 + 2x - 1 + 2$

$\longrightarrow\sf\textbf\:x^4 + 4x^2 + 2x - 1 + 2$

$\longrightarrow\sf\textbf\:x^4 + 4x^2 + 2x + 1 - 2$

$\longrightarrow\sf\textbf\:(x^2 + 1)^2 - 2$

Therefore, we can express the simplified forms of ${\sf{\textbf{f(x)}}}$ and ${\sf{\textbf{g(x)}}}$ as:

$\longrightarrow\sf\textbf\:f(x)\:=\:-x(x^3 - 2x + 1) - 4x + 3$

$\longrightarrow\sf\textbf\:g(x)\:=\:(x^2 + 1)^2 - 2$

[tex]\huge{\colorbox{black}{\textcolor{lime}{\textsf{\textbf{I\:hope\:this\:helps\:!}}}}}[/tex]

[tex]\begin{align}\colorbox{black}{\textcolor{white}{\underline{\underline{\sf{Please\: mark\: as\: brillinest !}}}}}\end{align}[/tex]

[tex]\textcolor{blue}{\small\textit{If you have any further questions, feel free to ask!}}[/tex]

[tex]{\bigstar{\underline{\boxed{\sf{\textbf{\color{red}{Sumit\:Roy}}}}}}}\\[/tex]

The probability of selecting two country songs followed by an R&B song is 0.0202.

The probability of selecting two country songs followed by an R&B song is determined below as follows:

The total number of songs in the playlist = 40

The probability of the first song being country = 15/40.

The probability of the second song also being country = 14/39

The probability of the third song being R&B =s 10/38

Therefore, the probability of selecting two country songs followed by an R&B song is:

Probability(country, country, R&B) = (15/40) x (14/39) x (10/38)

Probability(country, country, R&B)  = 0.0202

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Marissa needs to score 94% on test five to have a mean of 90% for all five tests.

What is average?

Let's look at the average formula in more detail in this part and use some examples to illustrate how it may be used. The following is an example of the average formula for a specific set of data or observations: Average = (Sum of Observations) ÷ (Total Numbers of Observations).

Let's use the formula for finding the mean:

mean = (sum of all scores) / (number of scores)

We know that Marissa has taken four tests and has a mean score of 89%, so:

(4 x 89) + x = 5 x 90

Simplifying:

356 + x = 450

x = 94

Therefore, Marissa needs to score 94% on test five to have a mean of 90% for all five tests.

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Lee spent a total of 33 hours working on all the lawns, as calculated by multiplying the number of lawns for each size category by the respective time required for mowing, trimming, and sweeping.

In order to determine the total number of hours Lee spent working on all the lawns, we need to calculate the time for each task separately. For the average size lawn, Lee takes 1 hour to mow and 2 hours to trim and sweep, totaling 3 hours per lawn. For the large size lawn, Lee takes 3 hours to mow and 3 hours to trim and sweep, totaling 6 hours per lawn.

Given that Lee mowed, trimmed, and swept 5 average size lawns and 3 large size lawns in one week, we can calculate the total hours as follows:

Total hours for average size lawns = 5 lawns * 3 hours/lawn = 15 hours

Total hours for large size lawns = 3 lawns * 6 hours/lawn = 18 hours

Therefore, the total hours Lee spent working on all the lawns is 15 hours + 18 hours = 33 hours.

In conclusion, Lee spent a total of 33 hours working on all the lawns, as calculated by multiplying the number of lawns for each size category by the respective time required for mowing, trimming, and sweeping.

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The total amount of money Victoria will have in her bank account after x years can be modeled by the function f(x) = 200 * (1.05)ˣ.

To model the total amount of money Victoria will have in her bank account after depositing her $200 and accruing interest over time, we can combine the two functions h(x) and s(x).

We can use the following formula to represent the total amount of money Victoria will have in her bank account after x years:

f(x) = h(x) + h(x) * s(x)

where h(x) represents the $200 that Victoria has saved at home, and h(x) * s(x) represents the amount of interest accrued on that $200 in x years according to the function s(x).

The justification for this formula is that the total amount of money Victoria will have in her bank account after x years is the sum of the initial amount of $200 and the interest accrued on that amount over x years.

The interest accrued can be calculated by multiplying the initial amount by the interest rate function s(x).

For example, if Victoria leaves her $200 in the bank for 5 years, the total amount of money she will have in her account can be calculated using the formula:

f(5) = h(5) + h(5) * s(5) = 200 + 200 * (1.05)⁴ ≈ $273.04

Therefore, the total amount of money Victoria will have in her bank account after x years can be modeled using the function f(x) = h(x) + h(x) * s(x).

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The solution to the composite function (f + g)(x) is: 8x + 7

Composite functions are said to occur when the output of one function is used as the input of another. If we have a function f and another function g, it means that the function fg(x), said as “ f of g of x”, is the composition of the two functions.

Now, we are given two functions as:

f(x) = x + 3

g(x) = 7x + 4

Thus, we can say that:

(f + g)(x) = f(x) + g(x)

= x + 3 + 7x + 4

= 8x + 7

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Therefore, the perimeter of the rectangle is 38 units that is option C.

A coordinate is a set of numbers or values that specifies the position or location of a point or object in a space. In mathematics, coordinates are used to describe the position of a point in a plane, a space or a higher-dimensional object. Coordinates can be represented by ordered pairs or tuples, where the first value corresponds to the position on the horizontal axis, and the second value corresponds to the position on the vertical axis.

Here,

First, we need to find the distance between points A and B to determine the length of the rectangle:

Distance AB = |yB - yA|

= |8 - 2|

= 6

Next, we need to find the distance between points B and C to determine the width of the rectangle:

Distance BC = |xC - xB|

= |6 - (-7)|

= 13

Since opposite sides of a rectangle are equal in length, we know that the distance between points A and D is also 13, and the distance between points C and D is also 6.

Therefore, the perimeter of the rectangle is:

Perimeter = 2 * (length + width)

Perimeter = 2 * (13 + 6)

Perimeter = 2 * 19

Perimeter = 38

So the answer is (C) 38.

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A graph of the solution to this inequality 3x - 4 > 11 is shown in the image attached below.

In Geometry, a number line simply refers to a type of graph that is composed of a graduated straight line, which typically comprises both negative and positive numerical values (numbers) that are located at equal intervals along its length.

In this scenario and exercise, we would determine the solution to the given inequality by solving for x as follows;

3x - 4 > 11

By adding the numerical value 4 to both sides of the inequality, we have the following:

3x - 4 + 4 > 11 + 4

3x > 15

x > 15/3

x > 5.

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