Shen does a weekly exercise program consisting of cardiovascular work and weight training. Each week, he exercises for at least 12 hours. He spends at most 9 hours on weight training. He spends at most 6 hours doing cardiovascular work. Let x denote the time (in hours) that Shen spends doing cardiovascular work. Let y denote the time (in hours) he spends on weight training. Shade the region corresponding to all values of x and y that satisfy these requirements.

PLEASE!!!!! GRAPH AND NUMBERS TOO!!!​

The graph of the compressed function is graph A.

Remember that for any function f(x), we define a horizontal compression of scale factor k (whre we must have k > 1) is defined as:

f(k*x)

Here we have f(4x), so this is an horizontal compression of scale factor 4.

Notice that the original function goes from x = -4 to x = 4

Then the compressed function must go from x = -4/4 = -1 to x = 4/4 = 1

Which is what we can see in graph A, so that is the correct option.

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The constraints of inequalities are 3x + 4y ≤ 640 and 75x + 60y ≤ 12900

Represent the types of cellphones with x and y

Using the problem statements, we have the following table of values

                         x             y       Available

Labor (hours)   3            4         640

Materials ($)     75         60       12900

From the above, we have the following constraints of inequalities:

3x + 4y ≤ 640              

75x + 60y ≤ 12900

The graph of the inequalities and the shaded region are added as an attachment

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Answer:

x  = -26/7

Step-by-step explanation:

Cancel terms that are in both the numerator and denominator

Multiply the numbers

Distribute

Subtract the numbers

Rearrange terms

Subtract the numbers

Multiply the numbers

Answer:

To solve this equation, we need to use the order of operations, which is PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

First, we need to simplify the expression inside the parentheses:

6 - 3/3(7x + 2) = 6(8 - 3)

6 - 1(7x + 2) = 6(5)

6 - 7x - 2 = 30

4 - 7x = 30

Next, we need to isolate the variable (x) on one side of the equation. We can do this by subtracting 4 from both sides:

4 - 7x - 4 = 30 - 4

-7x = 26

Finally, we can solve for x by dividing both sides by -7:

x = -26/7

Therefore, the solution to the equation is x = -26/7.

Answer:

The price of an ice cream sundae is $4.75.

Step-by-step explanation:

Let m be the cost of one milkshake and s be the cost of one ice cream sundae.

4m + 2s = 23.50 -------> 8m + 4s = 47.00

8m + 6s = 56.50 -------> 8m + 6s = 56.50

------------------------

2s = 9.50

s = 4.75

Answer:

Option A.

Step-by-step explanation:

A 30° - 60° - 90° triangle is a Right Triangle that has special side measures.

Let's summarize the sides in a ratio.

[tex]Short \ Leg: Long \ Leg: Hypotenuse\\x:x \sqrt{3} : 2x[/tex]

The short leg is just x.

The long leg is multiplied by [tex]\sqrt{3}[/tex].

The hypotenuse is double of the short leg.

For example, if the short leg is 2;

[tex]Short \ Leg = 2\\Long \ Leg = 2\sqrt{3} \\Hypotenuse = 4[/tex]

Let's look at the 4 options provided. We should check if the values of the sides match with a 30° - 60° - 90° triangle.

Option A has a short leg with the value of 5.

The long leg is correct, it's multiplied by [tex]\sqrt{3}[/tex].

The hypotenuse is correct, it's double of the short leg.

Option A is correct!

Option B has a short leg with the value of 5.

The long leg is correct, it's multiplied by [tex]\sqrt{3}[/tex].

The hypotenuse is incorrect, it's triple of the short leg.

Option B is incorrect.

Option C has a short leg with the value of 5.

The long leg is incorrect, it's multiplied by [tex]2\sqrt{3}[/tex].

The hypotenuse is correct, it's double of the short leg.

Option C is incorrect.

Option D has a short leg with the value of 10.

The long leg is incorrect, it's multiplied by [tex]\frac{1}{2} \sqrt{3}[/tex].

The hypotenuse is incorrect, it's [tex]1 \frac{1}{2}[/tex] of the short leg.

Option D is incorrect.

Our only 30° - 60° - 90° triangle is Option A.

Answer:

250 cubic cm

Step-by-step explanation:

Side length = x

[tex]2x^{2} + 4x(10) = 120[/tex]

[tex]x^{2} +2x - 30 =0[/tex]
After factorization, we will get (x+6) ( x-5) = 0

side length should be positive, so we take x to be 5.
Dimensions will be 5 x 5 x 10 = 250 cubic centimeters.

The mass of 5600 paper clips is 5.6 kilograms.

Here we use the unitary method to solve the problem. The unitary method is a mathematical technique used to solve problems.

It involves finding the value of one unit of a given quantity and then using that value to determine the value of other units of the same or different quantities.

Here we have

One paperclip has a mass of 1 gram.  

Mass of 1000 paperclips = 1 kilogram  

The mass of 1 paperclip in kilogram = 1/1000 = 0.001 kg

Similarly

Mass of 5600 paperclips = 0.001 kg × 5600 = 5.6 kg

Therefore,

The mass of 5600 paper clips is 5.6 kilograms.

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The distance from first base to dead center field in a certain baseball stadium is 338 feet.

Explanation:

The distance from first base to dead center field in a certain baseball stadium is 338 feet. Given,The distance from home plate to dead center field in a certain baseball stadium is 407 feet.A baseball diamond is a square with a distance from home plate to first base of 90 feet.

To find,How far is it from first base to dead center field?

Solution:Given that the distance from home plate to dead center field is 407 feet.The baseball diamond is a square with a distance from home plate to first base of 90 feet.Now we have to find the distance from first base to dead center field.We can find the distance by using the Pythagorean theorem which states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.Let us consider a right triangle ABC where AB represents the distance from home plate to first base, AC represents the distance from home plate to dead center field, and BC represents the distance from first base to dead center field.

As per the Pythagorean theorem, we have

AC² = AB² + BC²

Putting the values, we have

AC² = (90)² + BC²AC² = 8100 + BC²AC² - BC² = 8100

Taking the square root on both sides, we getAC = √(8100 + BC²)

Now we have AC = 407 ft,AB = 90 ftAC² = AB² + BC²407² = 90² + BC²BC² = 407² - 90²BC² = 165649BC = √165649BC = 407 ft - 90 ft

BC = 338 ft

Therefore, the distance from first base to dead center field in a certain baseball stadium is 338 feet.

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Answer:

Not remain

Step-by-step explanation:

The results of the composition of two functions are listed below:

Case 1:  

a) f[g(2)] = 3

b) f[g(x)] = 18 · x² - 60 · x + 51

c) g[f(x)] = 6 · x² - 2

d) g[g(x)] = 9 · x - 20

e) f[f(- 2)] = 163

Case 13: g[f(x)] = (√x + 2)² + 3, Dom {g[f(x)]} = [0, + ∞)

Case 15: g[f(x)] = (3√x + 1) / [27 · (√x)³], Dom {g[f(x)]} = (0, + ∞)

Case 17: g[f(x)] = 1 / [[(2 / x) + 4] - 4] = x / 2, Dom {g[f(x)]} = All real numbers.

Case 21: g[f(x)] = - 3 / (√2 - 4 · x), Dom {g[f(x)]} = All real numbers except x = √2 / 4.

In this problem we must determine, analyze and evaluate the composition of two functions, whose definition is shown below:

f ° g (x) = f [g (x)]

Now we proceed to determine the composition of functions:

Case 1: f(x) = 2 · x² + 1, g(x) = 3 · x - 5

a) f[g(2)] = 2 · (3 · 2 - 5)² + 1 = 2 · 1² + 1 = 2 + 1 = 3

b) f[g(x)] = 2 · (3 · x - 5)² + 1 = 2 · (9 · x² - 30 · x + 25) + 1 = 18 · x² - 60 · x + 51

c) g[f(x)] = 3 · (2 · x² + 1) - 5 = 6 · x² - 2

d) g[g(x)] = 3 · (3 · x - 5) - 5 = 9 · x - 20

e) f[f(- 2)] = 2 · [2 · (- 2)² + 1]² + 1 = 2 · (2 · 4 + 1)² + 1 = 2 · 9² + 1 = 162 + 1 = 163

Case 13: f(x) = √x + 2, g(x) = x² + 3

g[f(x)] = (√x + 2)² + 3

Dom {g[f(x)]} = [0, + ∞)

Case 15: f(x) = 3√x, g(x) = (x + 1) / x³

g[f(x)] = (3√x + 1) / (3√x)³ = (3√x + 1) / [27 · (√x)³]

Dom {g[f(x)]} = (0, + ∞)

Case 17: f(x) = 1 / (x - 4), g(x) = (2 / x) + 4

g[f(x)] = 1 / [[(2 / x) + 4] - 4] = x / 2

Dom {g[f(x)]} = All real numbers.

Case 21: f(x) = √2 - 4 · x, g(x) = - 3 / x

g[f(x)] = - 3 / (√2 - 4 · x)

Dom {g[f(x)]} = All real numbers except x = √2 / 4.

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6.2y - 3.7 = 6.2y + n n = -3.7 is the value we use to put this equal to and then solve for n. Hence, -3.7 is the value of p that equalises the two equations.

We need to simplify both equations and set them equal to one another in order to get the value of p that makes the expressions comparable.

Putting the left half of the equation first: Group like words to get (-y + 5.3) + (7.2y - 9) as -y + 7.2y - 9 + 5.3.

We will now put this equal to 6.2y + n and get n: 6.2y - 3.7 = 6.2y + n \sn = -3.7

Hence, -3.7 is the value of p that renders the equations equal.

A statement proving the equivalence of two mathematical expressions, sometimes incorporating one or more unknown variables, is known as an equation. Usually, an equal sign is used to denote it.

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The probabilities regarding a person spending are given as follows:

a) More than 28: 0.2033 = 20.33%.

b) Between 9 and 21: 0.3608 = 36.08%.

The z-score of a measure X of a variable that has mean symbolized by [tex]\mu[/tex] and standard deviation symbolized by [tex]\sigma[/tex] is obtained by the rule presented as follows:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The mean and the standard deviation for this problem are given as follows:

[tex]\mu = 23, \sigma = 6[/tex]

The probability of a person spending more than 28 is one subtracted by the p-value of Z when X = 28, hence:

Z = (28 - 23)/6

Z = 0.83

Z = 0.83 has a p-value of 0.7967.

1 - 0.7967 = 0.2033 = 20.33%.

The probability of spending between 9 and 21 is the p-value of Z when X = 21 subtracted by the p-value of Z when X = 9, hence:

Z = (21 - 23)/6

Z = -0.33

Z = -0.33 has a p-value of 0.3707.

Z = (9 - 23)/6

Z = -2.33

Z = -2.33 has a p-value of 0.0099.

Hence:

0.3707 - 0.0099 = 0.3608 = 36.08%.

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The value of y for x ranging from -4 to 4 are 34, 19, 6, -3, -2, 1, 10, 23 and 38, respectively.

To find the values of y for the given range of x, we can substitute each value of x into the equation y = 2x² - x - 2 and simplify. The completed table of values is:

(The table is attached below)

The given relation is y = 2x² - x - 2. To create a table of values for this relation, we can substitute values of x in the given range of -4 to 4 and calculate the corresponding values of y using the equation. In the first table, we substitute values of x ranging from -4 to 0 and calculate the corresponding values of y. In the second table, we substitute values of x ranging from 0 to 4 and calculate the corresponding values of y. These tables show how the values of y change as x changes within the given range, and provide a way to graph the relation as well.

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the number that is not part of the solution set is A) 0.

How to find and what does variable mean?

To solve the inequality x + 2 ≥ 3, we need to isolate the variable x.

x + 2 ≥ 3

Subtract 2 from both sides:

x ≥ 1

This means that any value of x that is greater than or equal to 1 is part of the solution set.

To check which of the given numbers is not part of the solution set, we need to substitute each of them in the inequality and see if it is true or false.

A) 0 + 2 ≥ 3 --> 2 ≥ 3 (False)

B) 2 + 2 ≥ 3 --> 4 ≥ 3 (True)

C) 3 + 2 ≥ 3 --> 5 ≥ 3 (True)

D) 6 + 2 ≥ 3 --> 8 ≥ 3 (True)

Therefore, the number that is not part of the solution set is A) 0.

In mathematics, a variable is a symbol or letter that represents a value or a quantity that can vary or change. It is often used to represent unknown or undefined values or quantities, and is commonly denoted by letters such as x, y, z, a, b, and c.

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The most likely measurement of the distance from the earth to the moon would be 1.3 x 10^8 ft.

Therefore the answer is A)1. 3 x 10⁸ ft.

This is because the distance from the earth to the moon is approximately 238,900 miles, which is equivalent to approximately 1.3 x 10^8 feet. Options B, C, and D are all significantly larger or smaller than this value and do not reflect the actual distance from the earth to the moon.

In general, measurements of distance can be expressed in a variety of units, such as feet, meters, or miles. It's important to use the correct unit when making calculations or comparisons to ensure that the results are accurate and meaningful. When dealing with very large or very small distances, scientific notation can be a useful way to express the measurement in a compact and standardized form.

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--The question is incomplete, answering to the question below--

"which value would be the most likely measurement of the distance from the earth to the moon?

A)1. 3 x 10⁸ ft.

B)1. 3 x 10⁻⁹ ft.

C)1. 3 x 10¹⁰⁰ ft.

D)1. 3 x 10² ft."

Answer: 20%

Step-by-step explanation:

150,000 - 125,000 = 25,000

20 percent of 125,000 = 25k

On solving the inequalities 2(5-x) > 6 and 22 > 2(9+x) the correct comparison is "The inequalities have the same solutions."

What is an inequality?

In Algebra, an inequality is a mathematical statement that uses the inequality symbol to illustrate the relationship between two expressions. An inequality symbol has non-equal expressions on both sides. It indicates that the phrase on the left should be bigger or smaller than the expression on the right, or vice versa.

To solve the inequality 2(5-x) > 6, we can simplify as follows -

2(5-x) > 6

10 - 2x > 6

-2x > 6 - 10

-2x > -4

x < 2

So the solution to this inequality is x < 2.

To solve the inequality 22 > 2(9+x), we can simplify as follows -

22 > 2(9+x)

22 > 18 + 2x

4 > 2x

2 > x

So the solution to this inequality is x < 2.

Both inequalities have the same solution, x < 2.

Therefore, the correct comparison of solutions is option D.

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The evaluation of the expression "twelve more than the product of 5 and a number x" evaluated at x=20 is equal to 112.

The expression "the product of 5 and a number x" can be written as 5x. Then, "twelve more than the product of 5 and a number x" is 5x + 12.

The expression "12 more than the product of 5 and x" can be written as "5x + 12." If x=20, then the expression evaluates to 5(20) + 12 = 100 + 12 = 112. Therefore, when x=20, "twelve more than the product of 5 and a number x" is equal to 112.

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Complete Question:

twelve more than the product of 5 and a number x? x is 20.

The two events are independent when one event has no effect on another event.

Question:2 ( 15 marks) An urn contains seven black balls and five white balls. Three balls are randomly awn from the urn without replacement and let X denote the number of black balls drawn. Then e three balls are put back to the urn. Next, four balls are randomly drawn from the urn without placement and let Y denote the number of black balls drawn. Let A=X+Y and B=X−Y . a. Find Cov(A,B) . b. Find Var(A+B) . c. Are A and B independent? Why?Solution:Given,An urn contains seven black balls and five white balls. Three balls are randomly drawn from the urn without replacement and let X denote the number of black balls drawn.Then these three balls are put back to the urn.Next, four balls are randomly drawn from the urn without placement and let Y denote the number of black balls drawn.Let A=X+Y and B=X−Y.a. Cov(A,B)Cov(A, B) = Cov(X+Y, X-Y)= Cov(X,X) - Cov(X,Y) + Cov(Y,X) - Cov(Y,Y) = Var(X) - Cov(X,Y) - Cov(Y,X) + Var(Y)Also, Var(X) = E(X^2) - [E(X)]^2=E(X)(E(X)+1)-E(X)^2 = E(X) + E(X)^2 - E(X)^2 = E(X) = 3.5Simillarly, Var(Y) = E(Y) + E(Y)^2 - E(Y)^2 = E(Y) = 2The probability of choosing a black ball from the urn P(X=1) = (7/12), The probability of choosing a black ball from the urn P(Y=1) = (7/12)Cov(X, Y) = E(XY) - E(X)E(Y) E(XY) = P(X=1,Y=1) + P(X=0,Y=0) = (7/12) * (6/11) * (5/10) + (5/12) * (4/11) * (3/10) = 21/110Cov(X, Y) = E(XY) - E(X)E(Y) = (21/110) - 3.5*2 = -14/110=-0.1272Thus, Cov(A, B) = 3.5 + 2 + 2 - 14/110 = 54/55=0.9818b. Var(A+B)Var(A+B) = Var(X+Y+X-Y) = Var(2X) + Var(2Y) = 4Var(X) + 4Var(Y) = 4(3.5) + 4(2) = 16Thus, Var(A+B) = 16.c. Are A and B independent? Why?A and B are not independent because Cov(A,B) ≠ 0, where Cov(A,B) = 54/55 ≠ 0.Note: In statistics, independence is a condition in which two events (A and B) are unrelated to each other. If the probability of an event A is not affected by the occurrence of another event B, these events are independent. That means, two events are independent when one event has no effect on another event.

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The expression 8* is equivalent to both 32 and 32².Expression 8* is an algebraic expression that can be expanded to 8*1 = 8. Thus, 8* is equivalent to 8.

An algebraic expression is a combination of terms, variables and operators that when simplified, evaluates to a numerical value. It is usually written in the form of equations and inequalities, where the terms represent numbers and variables represent unknown values.

When multiplied by 1, 8* is equivalent to 32. This is because 8*1 = 8 and 32*1 = 32. So, 8* = 32.

Expression 8* is also equivalent to 32². This is because 8*1 = 8 and 32² is 8 multiplied by itself. So 8* = 32².

The expressions A. (²/²), B. 8.8x+1, and F. 8-8-1 are not equivalent to 8*.

Expression A. (²/²) is not equivalent to 8*. This is because (²/²) is an undefined expression, while 8* is a defined expression.

Expression B. 8.8x+1 is not equivalent to 8*. This is because 8.8x+1 is an algebraic expression that includes the variable x, while 8* does not include any variables.

Expression F. 8-8-1 is not equivalent to 8*. This is because 8-8-1 is a subtraction expression, while 8* is a multiplication expression.

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Answer:

Step-by-step explanation:

3/70

The probability that the next customer pays with the debit card is 3/70 or 0.04.

Probability:

The probability of an event is a number that indicates the probability of the event occurring. Expressed as a number between 0 and 1 or as a percent sign between 0% and 100%.

Given that:

Last week there are 32 customer who paid cash

03 customers used debit card

35 customer used credit card.

Now,

The Total Number is = 32 + 3 + 35

                                   = 70

For cash buyer = 32/70 = 16/35

                         = 0.45

For Debit Card = 3/70

                         = 0.04

For Credit Card = 35/70 = 1/2

                          = 0.5

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Answer:

D

Step-by-step explanation:

every term of one expression gets multiplied with every term of the other expression.

(6x² - 3x - 6)(4x² + 5x + 4) =

= 6×4x²×x² + 6×5x²×x + 6×4x² - 3×4x×x² - 3×5x×x -

3×4x - 6×4x² - 6×5x - 6×4

3 terms × 3 terms = 9 terms.

now we combine similar factors for the 9 terms

24x⁴ + 30x³ + 24x² - 12x³ - 15x² - 12x - 24x² - 30x - 24

and now we combine similar terms

24x⁴ + 18x³ - 15x² - 42x - 24

The factored form of 196x²- y² is (14x + y)(14x - y).

A factored form is a parenthesized algebraic expression. In effect a factored form is a product of sums of products, or a sum of products of sums. Any logic function can be represented by a factored form, and any factored form is a representation of some logic function.

The slope intercept form of a straight line is one of the most common forms used to represent the equation of a line. The slope intercept formula can be used to find the equation of a line when given the slope of the straight line and the y-intercept. The formula is y=mx+b.

The expression 196x² - y² can be factored using the difference of squares formula, which states that:

a²- b² = (a + b)(a - b)

In this case, we have a = 14x and b = y, so we can write:

196x² - y² = (14x + y)(14x - y)

Therefore, the factored form of 196x²- y² is (14x + y)(14x - y).

The expression (14x + y)(14x - y) is the factored form of a quadratic expression and does not represent a linear equation that can be written in slope-intercept form.

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The new circle equation is: (x - 4)² + y² = 25

The x-intercepts are:  (9, 0) and (-1, 0)

You already have the answer for the first question, so let's look at b and c.

Here we have a circle equation:

x² + y² = 25

And now this circle is translated by the vector (4, 0) to make a new circle, so the new center will be at the point (4, 0).

That means that the new coordinates of the circle are:

(x - 4)² + (y - 0)² = 25

(x - 4)² + y² = 25

Now we want to get the intercepts of the x-axis, to get these we need to take y = 0.

then:

(x - 4)²  = 25

(x - 4) = ±5

x = 4 ± 5

The x-intercepts are (9, 0) and (-1, 0)

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Answer:

Farrah borrowed $155 from her brother and has paid back $15 so far. She plans to pay back $35 each month until the debt is paid off.

To determine the number of months it will take to pay off the debt, we need to solve the equation:

x * 35 + 15 = 155

where x is the number of months it will take to pay off the debt.

Simplifying the equation, we get:

x * 35 = 155 - 15

x * 35 = 140

x = 4

Therefore, it will take 4 months to pay off the debt.

Options that describe the number of months it will take to pay off the debt are:

Step-by-step explanation:

The probability of the number of defective bulbs in the sample greater than 15 is approximately 0.9177.Hence option A is correct.

Suppose that a factory produces light bulbs, and the percentage of defective bulbs is 3.5%. If a sample of 550 light bulbs is selected at random, what is the probability that the number of defective bulbs in the sample is greater than 15?We are required to calculate the probability of the number of defective bulbs in the sample greater than 15. Percentage of defective bulbs = 3.5%Number of light bulbs in the sample = 550Let X be the number of defective bulbs in the sample. We know that the probability distribution of X is a binomial probability distribution because the sample is selected randomly and the sample size is less than 10% of the total population,

which is considered as large. Let P (X > 15) be the probability of the number of defective bulbs greater than 15.Now, mean μ = np = 550 × 0.035 = 19.25 and standard deviation σ = √npq Where q = 1 − p = 1 − 0.035 = 0.965∴ σ = √npq = √550 × 0.035 × 0.965 = 3.05Therefore, Z = (15 − μ) / σ = (15 − 19.25) / 3.05 ≈ −1.39Using normal distribution, P (Z > −1.39) = 1 − P (Z ≤ −1.39)We get P (Z ≤ −1.39) = 0.0823Using the standard normal table or calculator, we get P (Z > −1.39) = 1 − 0.0823 = 0.9177Therefore, the probability of the number of defective bulbs in the sample greater than 15 is approximately 0.9177. Hence, option A is correct.

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The cost of the fabric needed to make the square table cloth is $3.75

Firstly, it's essential to understand that the cost of the fabric is calculated per square foot. This means that the more square feet of fabric you use, the higher the cost will be. Therefore, the first step is to measure the dimensions of the tablecloth.

For instance, if the tablecloth is 5 feet long and 3 feet wide, the total area would be 15 square feet.

Next, you can multiply the total square footage of the tablecloth by the cost of the fabric per square foot.

In this case, if the cost of the fabric is $0.25 per square foot, the cost of the fabric needed for the tablecloth would be

=> 15 (total square feet) x 0.25 (cost per square foot) = $3.75.

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The length of XZ is 5.5 m.

The length of side XZ is calculated by applying the following cosine and sine rule.

If the length of WY is 7 m, then ∠WYZ is calculated as follows;

cos Y = (z² + w² - y² ) / (2zw)

where;

cos Y = ( 7² + 5.1² - 3² ) / ( 2 x 7 x 5.1 )

cos Y = 0.9245

Y = cos⁻¹ (0.9245)

Y = 22.4⁰

The value of ∠WYX is calculated as follows;

cos Y = (x² + w² - y² ) / (2xw)

cos Y = ( 7² + 5² - 4.8² ) / ( 2 x 7 x 5)

cos Y = 0.728

Y = cos⁻¹ (0.728)

Y = 43.28⁰

The value of ∠ZYX = 43.28⁰ + 22.4⁰ = 65.68⁰

The length of XZ is calculated by using the following cosine rule.

|XZ|² = |XY|² + |ZY|² - (2 x |XY| x |XY|) cos Y

|XZ|² = 5² + 5.1²  -  (2 x 5 x 5.1 ) x cos (65.68)

|XZ|² = 30

|XZ| = √30

|XZ| = 5.5 m

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The x-coordinate where f'(x) = 0 and x is in the interval [0, pi] is:

x = π/6

What is derivative?

In calculus, the derivative of a function is a measure of how much the function changes as its input variable changes. More specifically, the derivative of a function f(x) at a particular value of x, denoted by f'(x), is defined as the limit of the ratio of the change in the function value to the change in the input variable as the change in the input variable approaches zero.

To find the x-coordinates where f'(x) = 0 for f(x) = 2x + sin(4x) in the interval [0, pi], we need to find the derivative of f(x) and set it equal to 0.

f(x) = 2x + sin(4x)

f'(x) = 2 + 4cos(4x)

Setting f'(x) equal to 0, we get:

2 + 4cos(4x) = 0

cos(4x) = -1/2

We know that cos(4x) = -1/2 has solutions at 4x = 2π/3 and 4x = 4π/3 (plus any multiple of 2π), because these are the solutions to cosθ = -1/2 in the interval [0,2π). So, we can write:

4x = 2π/3 or 4x = 4π/3

Solving for x in each equation, we get:

x = π/6 or x = π/3

However, we need to check that these solutions are in the interval [0, pi].

π/6 is in the interval [0, pi], but π/3 is not.

Therefore, the only x-coordinate where f'(x) = 0 and x is in the interval [0, pi] is:

x = π/6

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The values of the variables can be proved by solving the the equations to get;

8. y = 3

9. k = -2

10. w = 14

11. x = -9

An equation is a statement that indicates that two expressions are equivalent by joining them with the '=' sign.

The method used to prove the value of the variable is by solving the equations as follows;

8. (5·y - 1)/2 = 7

Therefore;

2 × 7 = 5·y - 1

14 = 5·y - 1

5·y = 14 + 1 = 15

y = 15/5 = 3

y = 3

9. 10·k - 4 = 2·k - 20

Therefore;

10·k - 2·k = 8·k = 4 - 20 = -16

8·k = -16

k = -16/8 = -2

Therefore;

k = -2

10. -8·(w + 1) = -5·(w + 10)

-8·w - 8 = -5·w - 50

Therefore;

-5·w + 8·w = 50 - 8 = 42

3·w = 42

w = 42/3 = 14

Therefore;

w = 14

11. 14 - 2·(x + 8) = 5·x - (3·x - 34)

Therefore;

14 - 2·x - 16 = 5·x - 3·x + 34

14 - 2·x - 16 = -2·x - 2

5·x - 3·x + 34 = 2·x + 34

Therefore;

-2·x - 2 = 2·x + 34

2·x + 2·x = -2 - 34 = -36

4·x = -36

x = -36/4 = -9

x = -9

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