FILL IN THE BLANK. Use part I of the Fundamental Theorem of Calculus to find the derivative of f(x) = x∫4 1/1+4t⁴ dt f'(x)=________

The y coordinate of the solution to this system of equations is -2

From the question, we have the following parameters that can be used in our computation:

3x+y=-2 x-2y=4

Express properly

So, we have

3x + y = -2

x - 2y = 4

Multiply the second equation by -3

so, we have the following representation

3x + y = -2

-3x + 6y = -12

Add the equations to eliminate x

7y = -14

Divide both sides by 7

y = -2

Hence, the value of y is -2

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The variance of the given data set is 18603.39.

To find the variance of the given data set {4, 44, 404, 244, 4, 74, 84, 64}, follow these steps:

Step 1: First, we need to find the mean of the data set:

Mean = (4 + 44 + 404 + 244 + 4 + 74 + 84 + 64) / 8 = 120.5

Step 2: Next, we calculate the deviation of each data point from the mean:

(4 - 120.5) = -116.5

(44 - 120.5) = -76.5

(404 - 120.5) = 283.5

(244 - 120.5) = 123.5

(4 - 120.5) = -116.5

(74 - 120.5) = -46.5

(84 - 120.5) = -36.5

(64 - 120.5) = -56.5

Step 3: Now, we square each deviation:

[tex](-116.5)^2 = 13556.25\\(-76.5)^2 = 5852.25\\(283.5)^2 = 80322.25\\(123.5)^2 = 15252.25\\(-116.5)^2 = 13556.25\\(-46.5)^2 = 2162.25 \\(-36.5)^2 = 1332.25\\(-56.5)^2 = 3192.25[/tex](-116.5)^2 = 13556.25

Step 4: We add up all the squared deviations:

13556.25 + 5852.25 + 80322.25 + 15252.25 + 13556.25 + 2162.25 + 1332.25 + 3192.25 = 130223.75

Step 5: We divide the sum of the squared deviations by the number of data points minus 1 to get the variance:

Variance = 130223.75 / 7 = 18603.39 (rounded to two decimal places)

Therefore, the variance of the data set is 18603.39.

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The sale price of the sofa can be represented by the equation p - 0.3p. This equation correctly represents the sale price after a 30% discount has been applied to the regular price.

The question is about the regular price of a sofa represented by p, and its sale price which is 30% off the regular price. You'd like to know if the statement "the sale price of the sofa can be represented by p-0.3p" is true.

Step 1: Understand the problem
The regular price of the sofa is represented by p. The sale price is 30% off the regular price.

Step 2: Represent the sale price
To find the sale price, we need to subtract the discount (30% of p) from the regular price (p).

Step 3: Calculate the discount
The discount can be calculated as 30% of p, which is 0.3 * p (or 0.3p).

Step 4: Determine the sale price
Now, subtract the discount from the regular price: p - 0.3p.

Step 5: Confirm the statement
The statement "the sale price of the sofa can be represented by p-0.3p" is indeed true.

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Based on the Root Test, the series Σ^n√14 is convergent.

Hi! To determine if the series Σ^n√14 is convergent or divergent, we need to analyze the terms involved. The series can be written as:

Σ (n√14)

This is a sum of terms, where each term is the n-th root of 14, and we want to find out if the sum converges or diverges as n goes to infinity.

In this case, the series is a type of p-series, where the terms follow the general form of 1/n^p. To be a convergent p-series, p must be greater than 1. Here, the terms are in the form of 14^(1/n), which can be rewritten as (14^(1))^(-n) or 14^(-n). This is not a p-series, as the exponent is not in the form of 1/n^p.

To further analyze the series, we can use the Divergence Test. If the limit of the terms as n goes to infinity is not equal to zero, then the series is divergent. So, let's find the limit:

lim (n → ∞) (14^(-n))

As n approaches infinity, the exponent -n becomes increasingly negative, and 14^(-n) approaches 0. However, the Divergence Test is inconclusive in this case, as it only confirms divergence if the limit is not equal to zero.

To determine convergence or divergence, we can use the Root Test. The Root Test states that if the limit of the n-th root of the absolute value of the terms as n goes to infinity is less than 1, then the series converges. Let's find the limit:

lim (n → ∞) |(14^(-n))|^(1/n)

This simplifies to:

lim (n → ∞) 14^(-1)

Since 14^(-1) is a constant value less than 1, the limit is less than 1.

Thus, based on the Root Test, the series Σ^n√14 is convergent.

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The measure of the angle ∠S, obtained using the law of cosines is about 59.4°

The law of cosines states that the square of the length of a side of a triangle is equivalent to the sum of the squares of the other two sides less the product of the length of the other two sides and the angle between them. Mathematically; a² = b² + c² - 2·b·c·cos(A)

Where;

a, b, and c = The length of the sides of the triangle

A = The angle between b and c

The lengths of the sides of the triangle, obtained from a similar triangle are;

s = 50 inches, t = 58 inches, and u = 27 inches

The measure of the angle ∠S is required

The angle ∠S is the angle facing TU or the side s

According to law of cosines, we get;

s² = t² + u² - 2·t·u·cos(∠S)

Therefore;

50² = 58² + 27² - 2 × 58 × 27 × cos(∠S)

2 × 58 × 27 × cos(∠S) = 58² + 27² - 50²

cos(∠S) = (58² + 27² - 50²)/(2 × 58 × 27)

∠S = cos((58² + 27² - 50²)/(2 × 58 × 27)) ≈ 59.4°

The measure of the angle ∠S is about 59.4°

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The amount of paint that he will need to paint all six surfaces of the sandbox is: 68 square feet

Since the image is a rectangular prism

The volume of the box can be obtained by using the formula:

Volume = l * b * h

The box has a dimension of 1ft x 4ft x 6ft

The volume of the box = 1 x 4 x 6 = 24 cubic feet

Therefore, the volume of sand needed to fill the box will be = 24 cubic feet of sand

The surface area of the box can be obtained using the formula:

2(lb + lh + bh)

= 2(1*4 + 1*6 + 4*6)

=2(4 + 6 + 24)

=2 (34)

= 68 square feet

Therefore a total surface area of 68 square feet needs to be painted

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Complete question is:

Your cousin is building a Sandbox for his daughter how much sand will he need to fill the Box? Explain. How much paint will he need to paint all six surface of the sandbox? Explain. 1ft 4ft 6ft not answer choices

Given the scenerio in the picture about corn, the statement that is true looking at a significance level of 5% is The result is not statistically significant which implies that spraying the corn plants with the new type of fertilizer does increase the growth rate.

Looking at the statement "The result is not statistically significant,"this means that the p-value (probability value) of the test was greater than 0.05. It could have 0.15 oe 0.2.

When a test is greater than 0.05 or 5 % significance level, it shows that the what is happening to the corn (increase in growth rate) could have been as a result of chance alone.

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Management should hire 25 additional vendors to maximize the number of sodas sold.

Let x be the number of additional vendors that management hires. Then the total number of vendors is 40 + x, and the yield per vendor is 200 - 4x (since the yield decreases by 4 for each additional vendor).

The total number of sodas sold is the product of the number of vendors and the yield per vendor:

Total sodas sold = (40 + x) * (200 - 4x)

To maximize the number of sodas sold, we take the derivative of this expression with respect to x and set it equal to zero:

d/dx [(40 + x) * (200 - 4x)] =

Expanding and simplifying, we get:

-8x² + 120x + 8000 = 0

Dividing both sides by -8, we get:

x² - 15x - 1000 = 0

Using the quadratic formula, we solve for x:

x = (15 ± sqrt(15² + 411000)) / 2

x = (15 ± 35) / 2

x = -10 or x = 25

Since we can't hire a negative number of vendors, the only sensible solution is x = 25. Therefore, management should hire 25 additional vendors to maximize the number of sodas sold.

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Simple Interest is equal to ($800) (2000 x 8 x 5) / 100. The simple interest earned is therefore $800.

The measure of cash returned or earned over a set period of time on a principal sum of money is referred to as interest. It is frequently stated as a share of the principal sum and it may be either simple or complicated. Compounding interest is computed on the principal amount as well as any accrued but unpaid interest, whereas simple interest is assessed just on the principal amount. Loans, investments, and bank deposits frequently include interest.

given

We can use the following calculation to determine the simple interest received for a $2000 principal at an 8% rate over a 5-year period:

S.I =  (Principal x Rate x Time)/100

In this instance, Principal is $2000, Rate is 8% annually, and Term is 5 years.

With these values entered into the formula, we obtain:

Simple Interest is equal to ($800) (2000 x 8 x 5) / 100. The simple interest earned is therefore $800.

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The value of y is 50, and the perimeter of triangle ABC is approximately 49.3 units.

To find the value of y in the coordinate of vertex B, we can use the formula for the area of a triangle given the coordinates of its vertices:

Area =[tex]\frac{ 1}{2}[/tex] * |(x1(y2-y3) + x2(y3-y1) + x3(y1-y2))|

Let's substitute the given values into the formula:

12 = [tex]\frac{ 1}{2}[/tex]* |(3(y-6) + 8(6-1) + 4(1-y))|

Simplifying the equation:

24 = |(3y - 18 + 40 + 4 - 4y)|

24 = |(-y + 26)|

Now, we can solve the equation by considering both the positive and negative values of the absolute expression:

-y + 26 = 24

-y = -2

y = 2

-y + 26 = -24

-y = -50

y = 50

So we have two possible values for y: y = 2 or y = 50.

To determine the correct value for y, we need to analyze the given information further. Since we know that triangle ABC is not an isosceles triangle (as the base lengths differ), we can eliminate the possibility of y = 2, leaving us with y = 50.

Now, let's calculate the perimeter of triangle ABC using the coordinates of its vertices:

AB = [tex]\sqrt((8 - 3)^2 + (y - 1)^2)[/tex]

BC = [tex]\sqrt((4 - 8)^2 + (6 - y)^2)[/tex]

CA = [tex]\sqrt((3 - 4)^2 + (1 - 6)^2)[/tex]

Perimeter = AB + BC + CA

Substituting the known values:

Perimeter = [tex]\sqrt((8 - 3)^2 + (50 - 1)^2) + \sqrt((4 - 8)^2 + (6 - 50)^2) + \sqrt((3 - 4)^2 + (1 - 6)^2)[/tex]

Calculating each term:

Perimeter = [tex]\sqrt(25 + 2401) + \sqrt(16 + 2025) + \sqrt(1 + 25)[/tex]

Perimeter = [tex]\sqrt(2426) + \sqrt(2041) + \sqrt(26)[/tex]

Rounding the perimeter to the nearest tenth of a unit:

Perimeter ≈ 49.3 units

Therefore, the value of y is 50, and the perimeter of triangle ABC is approximately 49.3 units.

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Ms. Summers gives each child 1/16 gallon of milk.

The amount of milk that Ms. Summers gives to each child can be represented by the following expression:

(1/4 gallon of milk - 1/8 gallon of milk) / 2

This expression represents the amount of milk that remains after Ms. Summers drinks 1/8 gallon of milk, divided equally between her two children.

Simplifying the expression, we get:

(2/8 - 1/8) / 2 = 1/8 / 2 = 1/16

Therefore, Ms. Summers gives each child 1/16 gallon of milk.

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The maximum value of f(x, y) = x^y on the unit circle can be found using the constraint x^2 + y^2 = 1, which defines the unit circle. To solve this, we can use the method of Lagrange multipliers.

Let g(x, y) = x^2 + y^2 - 1. Then, the gradient of f(x, y) and the gradient of g(x, y) should be proportional:
∇f(x, y) = λ∇g(x, y)

Calculating the gradients:
∇f(x, y) = (yx^(y-1), x^y * ln(x))
∇g(x, y) = (2x, 2y)

Equating the components and dividing the equations, we get:
y * x^(y-1) / 2x = x^y * ln(x) / 2y

Simplifying, we obtain:
ln(x) = y

Now, using the constraint x^2 + y^2 = 1, we can substitute y with ln(x) and solve for x:
x^2 + (ln(x))^2 = 1

Numerically solving this equation, we get x ≈ 0.90097 and y ≈ ln(0.90097) ≈ -0.10536. Since we are only interested in positive values of x and y, this is the only solution in our domain. Now, we can find the maximum value of f(x, y):
f_max = f(0.90097, -0.10536) ≈ 0.79307

So the maximum value of f(x, y) on the unit circle is approximately 0.79307.

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The cost of a small cup of coffee is $2.97, and the cost of a piece of banana bread is $5.13.

Let x represent the cost of a small coffee and y represent the cost of a piece of banana bread. We know:

Cost of muffin: $3.51

y = x + $2.16

5(x + $3.51) + 2(x + y) = $48.60

Substitute y with x + $2.16:

5(x + $3.51) + 2(x + (x + $2.16)) = $48.60

Solve for x:

9x + $21.87 = $48.60

9x = $26.73

x = $2.97

Find y:

y = x + $2.16

y = $2.97 + $2.16

y = $5.13

A slice of banana bread costs $5.13, while a small coffee costs $2.97.

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Answer: 0= -5

Step-by-step explanation:

Number of cars that coasted more than 10.75 feet = 1

To solve this problem, you need to compare the distance each car coasted to 10.75 feet, which is the threshold for determining whether a car coasted more or less than 10.75 feet.

Manuel's car coasted 10 feet, which is less than 10.75 feet, so it did not coast more than 10.75 feet.

Richard's car coasted 10.5 feet, which is also less than 10.75 feet, so it did not coast more than 10.75 feet either.

Diego's car coasted 102 feet, which is more than 10.75 feet. Therefore, only one car coasted more than 10.75 feet, and the answer is 1.

So the answer is:

Number of cars that coasted more than 10.75 feet = 1

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The radius of the circular fountain is approximately 17 feet.

The formula for the volume of a circular fountain is given by V = πr^2h, where V is the volume, r is the radius, and h is the depth. In this case, we are given that the depth of the fountain is 3 feet and the volume is 1800 cubic feet. So we can plug in these values into the formula and solve for r as follows:

1800 = πr^2(3)

Simplifying this equation, we get:

r^2 = 600/π

Taking the square root of both sides, we get:

r = sqrt(600/π)

Using a calculator to approximate the value of sqrt(600/π), we get:

r ≈ 17

Therefore, the radius of the circular fountain is approximately 17 feet when rounded to the nearest whole number.

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The predicted exam grade for 1 hour of studying is 66. Therefore the residual for 1 hours of studying can be calculated as Residual = Actual Exam Grade - 66.

To find the residual for 1 hour of studying, we first need to determine the predicted exam grade and compare it to the actual exam grade for that specific data point.

Using the line of best fit equation y = 6x + 60, we can find the predicted exam grade for 1 hour of studying:

y = 6(1) + 60
y = 6 + 60
y = 66

So, the predicted exam grade for 1 hour of studying is 66. To calculate the residual, you need the actual exam grade for 1 hour of studying. If that information is not provided, the residual cannot be calculated. If the actual grade is provided, subtract the predicted grade from the actual grade to find the residual:

Residual = Actual Exam Grade - Predicted Exam Grade

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1) Leke pay $4.938 for his meal after a 40% discount.

2) -5 belongs to integers, rational numbers, and real number sets.

3) Target sells the shirt for $5.6 after they mark it up for 40%

1) Bill = 8.23 , discount coupon = 40 %

The total amount paid by leke after the discount coupon = 8.23 - (40 % of 8.23)

Total amount = 8.23 - ( 8.23 × 40/100 )

Total amount = 8.23 - 3.292

The total amount paid by Leke is $4.938

2) -5 is an integer because it is a negative whole number.

-5 is a rational number because it can be written in the form of a fraction.

-5 is a real number because it is a rational number .

-5 belongs to integers, rational numbers, and real number sets.

3) Shirt price = $4 , price increased = 40%

selling price = 4 + (40% of 4)

selling price = 4 + (4 × 40/100)

selling price = 4 + 1.6

Target sell shirt for $5.6

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Check the picture below.

[tex]\textit{using the pythagorean theorem} \\\\ a^2+o^2=c^2\implies o=\sqrt{c^2 - a^2} \end{array} \qquad \begin{cases} c=\stackrel{hypotenuse}{13}\\ a=\stackrel{adjacent}{5}\\ o=\stackrel{opposite}{h} \end{cases} \\\\\\ h=\sqrt{ 13^2 - 5^2}\implies h=\sqrt{ 169 - 25 } \implies h=\sqrt{ 144 }\implies h=12 \\\\[-0.35em] ~\dotfill[/tex]

[tex]\textit{volume of a pyramid}\\\\ V=\cfrac{Bh}{3} ~~ \begin{cases} B=\stackrel{base's}{area}\\ h=height\\[-0.5em] \hrulefill\\ B=\stackrel{10\times 10}{100}\\ h=12 \end{cases}\implies V=\cfrac{(100)(12)}{3}\implies V=400~in^3[/tex]

WHI Health Fund pays the most commission on the purchase of 100 shares with a commission of $96.00. Thus, option d is correct.

Funds offer price for  Upton Group = $18.96 - $18.47

Funds offer price for Green Energy fund = $18.01 - $17.29

Funds offer price for TJH Small-Cap fund = $19.05 - $18.43

Funds offer price for WHI Health fund = $20.96 - $20.00

To calculate the commission on purchasing shares, we need to find the allowance between the price ranges and then multiply the value by 100.

For the Upton Group fund, Commission = (Offer price - NAV) * 100

= ($18.96 - $18.47) * 100

= $49.00

For the Green Energy fund, Commission = (Offer price - NAV) * 100

= ($18.01 - $17.29) * 100

= $72.00

For the TJH Small-Cap fund, Commission = (Offer price - NAV) * 100

= ($19.05 - $18.43) * 100

= $62.00

For the WHI Health fund, Commission = (Offer price - NAV) * 100

= ($20.96 - $20.00) * 100

= $96.00

Therefore, we can conclude that the WHI Health fund pays the most commission of $96.00.

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The quantity that gives maximum profit is approximately 111.55 units.

To find the quantity that gives maximum profit, we need to first find the revenue function and then the profit function.

The revenue function is given by:

R(x) = xp = x(120,000 - 150x)

The profit function is given by:

P(x) = R(x) - C(x) = x(120,000 - 150x) - (700/x + 300x + x²)

To find the quantity that gives maximum profit, we need to find the derivative of the profit function and set it equal to zero:

P'(x) = 120,000 - 300x - 700/x² - 2x

Setting P'(x) equal to zero and solving for x, we get:

120,000 - 300x - 700/x² - 2x = 0

Multiplying both sides by x^2, we get:

120,000x² - 300x³ - 700 - 2x³ = 0

Simplifying, we get:

300x³ + 2x³ - 120,000x² - 700 = 0

Dividing both sides by 2, we get:

151x³ - 60,000x² - 350 = 0

Using a graphing calculator or numerical methods, we can find that the real root of this equation is approximately x = 111.55.

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A sample size of 40 attorneys is needed to estimate the mean IQ with 98% confidence and a margin of error of 3 IQ points.

To find the sample size, we use the formula:

n = (z*σ/E)²

where n is the sample size, z is the z-score for the desired level of confidence given as 98% , σ is the population standard deviation given as 15, and E is the margin of error given as 3 IQ points.

using the above values, we get:

n = (2.33*15/3)²

n = 39.05

Therefore, we need a sample size of at least 40 attorneys to estimate the mean IQ with 98% confidence and a margin of error of 3 IQ points.

Whether this sample size is practical or not depends on various factors, such as the availability of attorneys with the desired characteristics, the cost and time required to collect the data, and the resources available for analysis. In general, a sample size of 40 is considered moderate to large for many applications, and it may be feasible depending on the specific context.

A sample size of 40 attorneys is needed to estimate the mean IQ with 98% confidence and a margin of error of 3 IQ points.

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The length of side x in simplest radical form with a rational denominator is x√3.

To find the length of side x in simplest radical form with a rational denominator given x√3, some steps need to be followed.

Steps are:
1. Identify the radical: In this case, it is √3.

2. Identify the denominator: To rationalize the denominator, we want to eliminate the radical from the denominator. Since the given expression has x√3, the denominator we need to rationalize is 1.

3. Rationalize the denominator: To do this, multiply the expression by a value that will cancel out the radical in the denominator without changing the value of the expression. Since our denominator is 1, we need to multiply the expression by √3/√3.

4. Multiply the expression: (x√3) * (√3/√3) = x√3 * √3 = x(√3)^2 = x(3).

5. Simplify the expression: x(3) = 3x.

So, the length of side x in simplest radical form with a rational denominator is 3x.

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Answer:

We can use the Pythagorean theorem to find the length of the third side of the triangle ABC:

AB^2 = AC^2 + BC^2

(29)½^2 = 5^2 + 2^2

29 = 25 + 4

29 = 29

So the triangle is a right triangle with angle A being the angle opposite the side AC. Therefore, we can use the tangent function to find tan A:

tan A = opposite/adjacent = AC/BC = 5/2

So the exact value of tan A is 5/2.

Account amount after 7 years will be approximately:
(a) $31,950.42 when compounded annually
(b) $32,166.25 when compounded semiannually

We'll be using the compound interest formula to find the amount in each account for both (a) annual compounding and (b) semiannual compounding.

The compound interest formula is: A = P(1 + r/ⁿ)ⁿᵃ

Where:
A = the future amount in the account
P = the principal (initial investment)
r = Annual interest rate
n = Interest is compounded per year in numbers
a = the number of years

(a) Annual Compounding:
In this case n = 1.

P = $24,000
r = 4% = 0.04
n = 1
t = 7

A = 24000(1 + 0.04/1)¹ˣ⁷
A = 24000(1 + 0.04)⁷
A = 24000(1.04)⁷
A ≈ $31,950.42

(b) Semiannual Compounding:
For semiannual compounding, the interest is compounded twice a year, so n = 2.

P = $24,000
r = 4% = 0.04
n = 2
t = 7

A = 24000(1 + 0.04/2)²ˣ⁷
A = 24000(1 + 0.02)¹⁴
A ≈ $32,166.25

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In the given problem, solving systematically, sales should be $6,000 during a week that $1,200 was spent on advertising.

If sales vary directly with the amount of money spent on advertising, it means that the ratio of sales to advertising spending is constant. We can use this ratio to find out what sales should be during a week that $1200 was spent on advertising.

Let the ratio of sales to advertising spending be denoted by k. Then, we have:

k = sales / advertising spending

From the information given, we know that:

k = 10,000 / 2,000 = 5

This means that for every dollar spent on advertising, $5 in sales are generated.

To find out what sales should be during a week that $1200 was spent on advertising, we can use the ratio k:

sales = k * advertising spending

sales = 5 * 1200

sales = 6000

Therefore, sales should be $6,000 during a week that $1,200 was spent on advertising.

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The standard deviation of the sampling distribution is 2.53. It would not be appropriate to use a normal distribution to model the sampling distribution.

(a) When dealing with a sampling distribution, the mean of the sampling distribution (μ_x) is equal to the mean of the population (μ). In this case, the mean of the population is 65. Therefore, the mean of the sampling distribution is also 65.

To calculate the standard deviation of the sampling distribution (σ_x), you will use the following formula: σ_x = σ / √n, where σ is the standard deviation of the population and n is the sample size. In this case, the standard deviation of the population is 8 and the sample size is 10. So the standard deviation of the sampling distribution is:

σ_x = 8 / √10 ≈ 2.53

(b) The Central Limit Theorem states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution, provided that the population is not heavily skewed or has extreme outliers. Since the scores are extremely skewed left and the sample size is only 10, it would not be appropriate to use a normal distribution to model the sampling distribution. A larger sample size would be needed to use a normal distribution model.

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Answer:

[tex]40 + 37 + 30 + 40 + 39 + 41 + 38 + n = 265 + n[/tex]

1)

[tex] \frac{265 + n}{8} = 43[/tex]

[tex]265 + n = 344[/tex]

[tex]n = 79[/tex]

2)

[tex] \frac{265 + n}{8} = 40[/tex]

[tex]265 + n = 320[/tex]

[tex]n = 55[/tex]

3)

[tex] \frac{265 + n}{8} = 38[/tex]

[tex]265 + n = 304[/tex]

[tex]n = 39[/tex]

The true statement is that if the quadrilateral is not a rectangle, it can  still be a parallelogram

The statement in the question is given as

A quadrilateral has two consecutive right angles

By definition, the parallelograms that have two consecutive right angles are rectangles and squares

This is because all the four angles in a rectangle and a square are right angles

Using the above as a guide, we can conclude that the quadrilateral can still be a parallelogram if the quadrilateral has two consecutive right angles and if the quadrilateral is not a rectangle

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The maximum profit is $3,900 if the marginal cost and marginal revenue are given by C'(x) = 20+ x/20 and R'(x) = 40 and fixed cost is $100.00.

In order to determine the maximum profit, we need to find the quantity (x) where the marginal cost (C'(x)) equals the marginal revenue (R'(x)). This is because when these two values are equal, we are maximizing the profit. The given functions are:

C'(x) = 20 + x/20
R'(x) = 40

First, set the marginal cost equal to the marginal revenue:

20 + x/20 = 40

Now, we need to solve for x:

x/20 = 40 - 20
x/20 = 20
x = 20 * 20
x = 400

So, the maximum profit occurs at a quantity of 400 units. To find the total cost (C(x)) and total revenue (R(x)), we need to integrate the marginal cost and marginal revenue functions:

C(x) = ∫(20 + x/20) dx = 20x + x^2/40 + C₁
R(x) = ∫40 dx = 40x + C₂

Since we have a fixed cost of $100, we know that C₁ = 100. We don't need C₂ to find the profit, as it will cancel out when we calculate it. Now, let's find the total cost and total revenue for 400 units:

C(400) = 20(400) + (400)^2/40 + 100 = 8000 + 4000 + 100 = 12100
R(400) = 40(400) = 16000

Finally, calculate the profit (P(x)):

P(x) = R(x) - C(x) = 16000 - 12100 = 3900

Therefore, the maximum profit is $3,900 when producing 400 units.

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