Write an inequality that represents the cost of each cookie.


At Cindy's Sweet Treats, cookies are packaged in boxes of 8. Depending on the cookie flavor, the most a box can cost is $16

3) The expression In(4x^5) - In(x^3) - In 4 can be simplified using the properties of logarithms. We know that ln(a) - ln(b) = ln(a/b) and ln(a^n) = n ln(a), so we can write:In(4x^5) - In(x^3) - In 4 = In[(4x^5)/(x^3)] - In 4= In(4x^2) - In 4= In(4x^2/4)= In(x^2)Thus, the simplified expression is In(x^2).4) To solve this problem, we need to use the formula for compound interest:A = P(1 + r/n)^(nt)where A is the final amount, P is the initial amount, r is the interest rate (as a decimal), n is the number of times interest is compounded per year, and t is the number of years.We want to find t when A = 3P and r = 0.02 (since the price of beef has inflated by 2%). We are told that interest is compounded biannually, so n = 2. Plugging in these values and solving for t, we get:3P = P(1 + 0.02/2)^(2t)3 = (1.01)^2tln(3) = ln(1.01^2t)ln(3) = 2t ln(1.01)t = ln(3) / (2 ln(1.01))Using a calculator, we find t ≈ 34.64 years. Therefore, it will take about 34.64 years for the price of beef to triple at a 2% biannual inflation rate.

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It will take approximately 110 years for the price of beef to triple when inflating 2% compounded biannually.

3) To simplify the expression In(4x^5) - In(x^3) - In(4), we will use the properties of logarithms:

- In(a) - In(b) = In(a/b)
- In(a^b) = b * In(a)

So, we can rewrite the expression as:

In(4x^5 / (x^3 * 4))

Now, we can simplify the expression inside the natural logarithm:

(4x^5) / (4x^3) = x^(5-3) = x^2

Thus, the simplified expression is:

In(x^2)

4) To find how long it will take for the price of beef to triple when inflating 2% compounded biannually, we can use the compound interest formula:

A = P(1 + r/n)^(nt)

where A is the final amount, P is the initial amount, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the time in years. In this case, we want the final amount to be triple the initial amount:

3P = P(1 + 0.02/2)^(2t)

To solve for t, we can divide both sides by P:

3 = (1 + 0.01)^(2t)

Now, take the natural logarithm of both sides and use the properties of logarithms:

ln(3) = ln((1 + 0.01)^(2t))
ln(3) = 2t * ln(1 + 0.01)

Finally, isolate t:

t = ln(3) / (2 * ln(1 + 0.01))

t ≈ 109.96

It will take approximately 110 years for the price of beef to triple when inflating 2% compounded biannually.

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To find the derivative of h(x) = 8x, we use the power rule, which states that the derivative of x^n is nx^(n-1). Applying this rule to h(x), we get h'(x) = 8.

To find the value of h'(4), we simply plug in x = 4 into our derivative expression: h'(4) = 8.

Therefore, the derivative of h(x) = 8x at x = 4 is h'(4) = 8.
To find the derivative of the function h(x) = 8x at x = 4, you can use the power rule for differentiation. The power rule states that if h(x) = x^n, then h'(x) = n * x^(n-1).

For h(x) = 8x, n = 1, so:

h'(x) = 1 * 8x^(1-1) = 8

Now, to find h'(4), just plug in x = 4:

h'(4) = 8

So, h'(4) = 8.

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Answer:

$1680

Step-by-step explanation:

PV = $1200

i = 5%

n = 8

Simple interest formula:

FV = PV (1 + i × n)

FV = 1200 (1 + 5% x 8)

FV = $1680

The point N(-5,-3) lies inside the circle centered at C(0,0) with radius √38.

Since the point M(0, √38) lies on the circle with center C(0,0), we can find the radius of the circle by finding the distance between M and C:

r = √[tex]((0 - 0)^2[/tex] + (√[tex]38 - 0)^2)[/tex] = √38

Now that we know the radius of the circle is √38, we can determine where the point N(-5,-3) lies relative to the circle. We can find the distance between N and the center of the circle:

d = √[tex]((-5 - 0)^2[/tex] + [tex](-3 - 0)^2)[/tex] = √34

Since the distance between N and the center of the circle is less than the radius of the circle, the point N is inside the circle. Therefore, N lies inside the circle centered at C(0,0) with radius √38.

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Answer:

16 peaches

Step-by-step explanation:

Let's break this down:

Total: 144 peaches

Number of boxes she has to fill evenly: 9

Question: How many peaches are able to fit into each box evenly?

144 peaches/9 boxes = 16 peaches

So, Ariana should pack 16 peaches into each box

Hope this helps :)

The equation of the tangent line at (6, f(6)) is y = 6x - 48.

a. The slope of the secant line joining (2, f(2)) and (7, f(7)) is:

slope = (f(7) - f(2)) / (7 - 2)

We can find f(7) and f(2) by plugging in x = 7 and x = 2 into the expression for f(x):

f(7) = 7² - 6(7) = 7

f(2) = 2² - 6(2) = -8

Substituting these values into the slope formula, we get:

slope = (7 - (-8)) / (7 - 2) = 3

Therefore, the slope of the secant line joining (2, f(2)) and (7, f(7)) is 3.

b. The slope of the secant line joining (6, f(6)) and (6 + h, f(6 + h)) is:

slope = (f(6 + h) - f(6)) / ((6 + h) - 6) = (f(6 + h) - f(6)) / h

We can find f(6) and f(6 + h) by plugging in x = 6 and x = 6 + h into the expression for f(x):

f(6) = 6² - 6(6) = -12

f(6 + h) = (6 + h)² - 6(6 + h) = h² - 6h + 36 - 36 - 6h = h² - 12h

Substituting these values into the slope formula, we get:

slope = (h² - 12h - (-12)) / h = h - 12

Therefore, the slope of the secant line joining (6, f(6)) and (6 + h, f(6 + h)) is h - 12.

c. The slope of the tangent line at (6, f(6)) is the derivative of f(x) at x = 6:

f'(x) = 2x - 6

f'(6) = 2(6) - 6 = 6

Therefore, the slope of the tangent line at (6, f(6)) is 6.

d. To find the equation of the tangent line at (6, f(6)), we use the point-slope form of a line:

y - f(6) = f'(6)(x - 6)

Substituting f(6) and f'(6) into this equation, we get:

y - (-12) = 6(x - 6)

Simplifying, we get:

y = 6x - 48

Therefore, the equation of the tangent line at (6, f(6)) is y = 6x - 48.

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The function used to add together the last employee's sales for the three months on the Employee Sales Summary sheet is: =SUM('Employee Sales January:Employee Sales March'!E5)

SUM(E16): This function adds up the values in cells E16 from the current sheet. If the last employee's sales for the three months are stored in cells E16, E17, and E18, then this function would correctly calculate the total.

E16+E16+E16: This expression adds up the value in cell E16 three times. If the last employee's sales for the three months are stored in cells E16, E17, and E18, then this expression would not calculate the total correctly.

SUM('Employee Sales October:Employee Sales December'!E16): This function adds up the values in cell E16 from all sheets between Employee Sales October and Employee Sales December (inclusive). If the last employee's sales for the three months are stored in cells E16, E17, and E18 on different sheets, then this function could be used to calculate the total.

SUM('Employee Sales January:Employee Sales March'!E5): This function adds up the values in cell E5 from all sheets between Employee Sales January and Employee Sales March (inclusive).

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Full Question: On the Employee Sales Summary sheet, the function used to add together the last employee's sales for the three months is ___________________.

Group of answer choices

The farmer should construct a rectangle that is twice as long as it is wide, with dimensions of 1216.56 m x 608.28 m, and should use 7301.36 m of fence to divide it in half parallel to the shorter side in order to minimize the cost of the fence.

To minimize the cost of the fence, the farmer should construct a rectangle that is twice as long as it is wide, with the dividing fence parallel to the shorter side. This will result in two identical rectangles each with an area of 375 000 m².

The perimeter of the rectangle can be calculated as follows:

P = 2L + 2W

where L is the length and W is the width.

Since the area of the rectangle is 750 000 m² and the length is twice the width, we can write:

L x W = 750 000

L = 2W

Substituting L = 2W into the equation for area, we get:

2W x W = 750 000

2W² = 750 000

W² = 375 000

W = 608.28 m

L = 2W = 1216.56 m

So the dimensions of the rectangle are 1216.56 m x 608.28 m.

The perimeter of each rectangle is:

P = 2L + 2W

P = 2(1216.56) + 2(608.28)

P = 3650.68 m

The total length of fence needed is twice the perimeter, since we are dividing the rectangle in half:

Total fence length = 2 x 3650.68 = 7301.36 m

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The surface area of the cylinder is 96cm²

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

Using the above as a guide, we have the following:

Surface area of the cylinder = The area around the cylinder + 2 * The area of the top

Substitute the known values in the above equation, so, we have the following representation

Surface area of the cylinder = 64 + 2 * 16

Evaluate

Surface area of the cylinder = 96

Hence, the surface area of the cylinder is 96cm²

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If the interest is compounded daily, the interest rate is 4.5%.

To determine the interest rate, we can use the compound interest formula:

A = P(1 + r/n)^(nt)

Where:

A = the final amount, $790

P = the principal, $350

r = the interest rate

n = the number of times the interest is compounded per year, in this case daily (n = 365)

t = the time period in years, 18

Substituting the values :

790 = 350(1 + r/365)³⁶⁵ˣ¹⁸

790 = 350(1 + r/365)⁶⁵⁷⁰

790/350 = (1 + r/365)⁶⁵⁷⁰

ln(790/350) = 6570 * ln (1 + r/365)

Using the property of logarithms that ln(1 + x) ~ x for small values of x, we can approximate the right-hand side as:

[ln(790/350)]/6570 = r/365

r = 0.045

r = 4.5%

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The range of values that represent the middle 95% of the data is from 20.18 to 47.82 or (20.18, 47.82).

A measurable speculation test is a strategy for factual deduction used to conclude whether the information within reach adequately support a specific speculation. We can make probabilistic statements about the parameters of the population thanks to hypothesis testing.

The middle 95% of a normal distribution is located within 1.96 standard deviations from the mean in both directions.

Therefore, the lower limit is:

34 - 1.96(7) = 20.18

And the upper limit is:

34 + 1.96(7) = 47.82

So the range of values that represent the middle 95% of the data is from 20.18 to 47.82 or (20.18, 47.82).

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The statement that is true is: "The range of the function is all real numbers less than or equal to 9

This quadratic function is indicated by a downward-opening parabola due to the negative coefficient of the squared term.

Found at coordinates (-2, 9), the vertex will be the highest point on this curved graph

Located on the x-axis at points (-5, 0) and (1, 0), each one serves as an intersection. Because of these intersections the following statement can be confidently said: "The range of the function consists of all actual numbers that are lesser or equal to 9."

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By setting up an inequality, the maximum number of vans that are needed to ferry 80 people are 7 vans.

To find the maximum number of vans needed to ferry 80 people using the given terms, let's set up an inequality. Let's use the variable "v" to represent the number of vans.

Since a van can ferry a maximum of 12 people, we can write the inequality as:

12v ≥ 80

Now, let's solve for "v":

Divide both sides of the inequality by 12.
v ≥ 80/12

Simplify the inequality.
v ≥ 6.67

Since we cannot have a fraction of a van, we need to round up to the nearest whole number:

v ≥ 7

Therefore, the maximum number of vans needed to ferry 80 people is 7 vans.

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Jackson's gross monthly income for 40 hours per week is approximately $3,201.70 and gross annual income s $38,480.

To find Jackson's gross monthly income, we first need to find his gross weekly income.

Jackson's hourly wage is $18.50, so his weekly gross income for 40 hours of work is:

40 hours/week x $18.50/hour = $740/week

Calculate annual income:

To determine the gross annual income, we need to consider how many weeks there are in a year. Assuming 52 weeks in a year:

Annual income = Weekly income * Number of weeks in a year

Annual income = $740 * 52 = $38,480

To find Jackson's gross monthly income, we can multiply his weekly gross income by the number of weeks in a month (approximately 4.33):

$740/week x 4.33 weeks/month ≈ $3,201.70/month

Therefore, Jackson's gross monthly income for 40 hours per week is approximately $3,201.70.

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Step-by-step explanation:

To estimate the difference between 5 1/2 and 3 5/9, we can first round the fractions to the nearest whole number or simpler fractions. In this case, we can round 1/2 to 1/2 and 5/9 to 1/2 as well. Now, we have:

5 1/2 − 3 1/2

Subtracting the whole numbers, we get:

5−3=2

Subtracting the fractions, we get:

1/2 − 1/2 = 0

So, the estimated difference is 2, which falls between 2 and 2 1/2. Therefore, Elise chose option D as the correct answer.

If Liquid a has a density of 1. 2 g/cm³, 150 cm of Liquid a is mixed with some of Liquid b to make Liquid c whose mass is 220 g and has a density of  1.1 g/cm³, then the density of liquid B is 0.8 g/cm³.

To find the density of liquid B, you can follow these steps:
1. Calculate the mass of liquid A using its density and volume:
Liquid A has a density of 1.2 g/cm³ and a volume of 150 cm³.

Mass of A = Density of A × Volume of A = 1.2 g/cm³ × 150 cm³ = 180 g

2. Calculate the mass of liquid B using the mass of liquid C and mass of liquid A:
Liquid C has a mass of 220 g.
Mass of B = Mass of C - Mass of A = 220 g - 180 g = 40 g

3. Calculate the volume of liquid C using its mass and density:
Liquid C has a density of 1.1 g/cm³.
Volume of C = Mass of C ÷ Density of C = 220 g ÷ 1.1 g/cm³ = 200 cm³

4. Calculate the volume of liquid B using the volume of liquid C and the volume of liquid A:
Volume of B = Volume of C - Volume of A = 200 cm³ - 150 cm³ = 50 cm³

5. Calculate the density of liquid B using it's mass and volume:
Density of B = Mass of B ÷ Volume of B = 40 g ÷ 50 cm³ = 0.8 g/cm³

So, the density of liquid B is 0.8 g/cm³.

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The inverse of the function y= (2/3)x^5-10 is y = [3/2(x + 10)]^1/5

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

y= (2/3)x^5-10

Swap the ocurrence of x and y

so, we have the following representation

x = (2/3)y^5-10

Next, we have

(2/3)y^5 = x + 10

This gives

y^5 = 3/2(x + 10)

Take the fifth root of both sides

y = [3/2(x + 10)]^1/5

Hence, the inverse function is y = [3/2(x + 10)]^1/5

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Monica's claim is incorrect. Even though her interest rate is higher, she will earn less interest, $10.20, after one year than Paul because her initial deposit is lower.

Paul will earn more interest of $13.20 because he deposited more money, even though his interest rate is slightly lower.

To determine who will earn more interest in one year, we shall calculate the interest earned by each person using the simple interest formula.

The formula for simple interest is:

I = P * r * t

where:

I = the interest earned

P = the principal (the amount deposited)

r = the interest rate (as a decimal)

t =s the time (in years)

For Monica, we are given:

P = $300

r = 0.034 (the interest rate is 3.4%)

t = 1 (the interest earned in one year)

Plugging these values, we have:

I = 300 * 0.034 * 1 = $10.20

So, Monica will earn $10.20 in interest after one year.

For Paul, we are given:

P = $400

r = 0.033 (interest rate = 3.3%)

t = 1 (interest earned in one year)

Plugging the values, we get:

I = 400 * 0.033 * 1 = $13.20

So, Paul will earn $13.20 in interest after one year.

Therefore, Monica's claim is incorrect. Monica will earn $10.20 in interest after one year while Paul will earn $13.20 in interest after one year.

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The likelihood of at least 12% of the sample being left-handed is approximately 0.007 or 0.7%.

Let X be the number of left-handed people in a sample of 800 individuals. Since the probability of a person being left-handed is 0.1, the probability of a person being right-handed is 0.9. Then, X follows a binomial distribution with n = 800 and p = 0.1.

P(X ≥ 0.12*800) = P(X ≥ 96)

where 96 is the smallest integer greater than or equal to 0.12*800.

[tex]P(X > =96)-P(X < 96)=1-[K=0 to 95](800 CHOOSE )(0.1^{k} (0.9)^{2} (800-k)[/tex]

This is the complement of the probability of getting less than 96 left-handed people in the sample. Using a calculator or statistical software, we can find that:

P(X ≥ 96) ≈ 0.007

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The total surplus of both towns together is represented by the polynomial [tex]3x^2 + 600x + 80,000.[/tex]

To find the total surplus of both towns together, we need to add the budget surplus of Alphaville and Betaville.

The budget surplus of Alphaville is represented by the polynomial [tex]2x^2 + 700x.[/tex]

The budget surplus of Betaville is represented by the polynomial x^2 - 100x + 80,000.

Therefore, the expression that represents the total surplus of both towns together is:

[tex](2x^2 + 700x) + (x^2 - 100x + 80,000)[/tex]

Simplifying this expression, we get:

[tex]3x^2 + 600x + 80,000[/tex]

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The number of paint bottles required is 49.716 bottles

Thus, 50 bottles are needed to paint the ornaments.

Surface area is the sum of all exterior surfaces on a three-dimensional object, representing the quantity of material that covers it. Computing an object's surface area entrails measuring each of its faces and then adding up their areas altogether.

If we take, for instance, a cube, its surface area would be calculated by multiplying the measurement of one face width by another and then multiplying this value by six (each cube has six sides). The units applied to measure surface area are usually in square feet or square centimeters.

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The corresponding measure of the parameters are;

i. 1ml = 0. 001 liter a.

ii. 1000ml = 1 liter b.

iii. 1 cl = 0. 01 liter c.

iv. 10dcl = 1 liter d.

v. 1cl = 100ml e.

v. 0. 01 cl = 1ml f.

To convert the factors, we need to know the following conversion rates.

We have;

1 milliliter = 0. 001 liter

1 centiliter = 0. 01 liter

1 deciliter = 0. 1 liter

1 cubic centimeter = 1 millimeter

Hence, the sizes are determined by the corresponding factor.

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Complete question:

Convert the following to their equivalent measurement for each letter

i. 1 ml = a liters

ii) b ml = 1 liters

iii) 1 cl = c liters

iv)d cl = 1 liters

v) 1 cl = e ml

vi) f cl = 1 ml

The measure of x in the intersected chord is 16.

If two chords intersect in a circle, then the measure of the angle formed is one half the sum of the measure of the arcs intercepted by the angle and its vertical angle.

Using the chord intersection angle theorem,

5x - 7  = 1 / 2 (119 + 27)

5x - 7 = 1 / 2 (146)

5x - 7 = 73

add 7 to both sides of the equation

5x - 7 = 73

5x - 7 + 7 = 73 + 7

5x = 80

divide both sides of the equation by  5

x = 80 / 5

x = 16

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The statement''El 91 no es un número primo porque tiene más divisores que el 1 y el 91''  is true because  91 is not a prime number.

A prime number is a positive integer that has only two divisors, 1 and itself. To check if 91 is a prime number, we need to find its divisors. We can start by dividing 91 by 2, but we find that 2 is not a divisor of 91. Next, we can try dividing it by 3, and we get 30 with a remainder of 1. This means that 3 is not a divisor of 91 either.

We continue dividing by 4, 5, 6, and so on until we reach 13, which gives us 7 as a quotient and 0 as a remainder. Therefore, the divisors of 91 are 1, 7, 13, and 91, which means that 91 is not a prime number because it has more than two divisors. Hence, the statement is true.

91 ÷ 2 = 45 r 1

91 ÷ 3 = 30 r 1

91 ÷ 4 = 22 r 3

91 ÷ 5 = 18 r 1

91 ÷ 6 = 15 r 1

91 ÷ 7 = 13 r 0

Since 91 has more than two divisors, it is not a prime number.

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If you charge more than your limit on your credit card, you will likely face several consequences, including:

1. Over-limit fees: Many credit card companies charge a fee if you exceed your credit limit. This fee can vary depending on the issuer, but it's typically around $25 to $35.

2. Increased interest rates: If you go over your credit limit, your credit card company may increase your interest rate, making it more expensive to carry a balance on your card.

3. Lower credit score: Exceeding your credit limit can negatively impact your credit score, as it's an indication of risky financial behavior.

4. Reduced credit availability: Your credit card company may reduce your credit limit if you consistently go over the limit, making it harder for you to access credit in the future.

5. Declined transactions: If you're significantly over your limit, your credit card company may start declining transactions to prevent further over-limit spending.

To avoid these consequences, it's important to monitor your spending and ensure you stay within your credit limit. If you find yourself consistently approaching your limit, consider requesting a credit limit increase or using other methods to manage your spending.

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The length of AC in triangle ABC is 24 inches.

In triangle ABC, let point D be on side AC such that AB = BD = DC = 12 inches. We are given that the measure of angle BDC is twice the measure of angle ABD.

To find the length of AC, we can use the Law of Cosines. This law relates the lengths of the sides of a triangle to the cosine of one of its angles. In this case, we can apply the Law of Cosines to triangle ABD to find the length of AD:

AD^2 = AB^2 + BD^2 - 2 * AB * BD * cos(ABD)

Since AB = BD = 12 inches, we have:

AD^2 = 12^2 + 12^2 - 2 * 12 * 12 * cos(ABD)

AD^2 = 288 - 288 * cos(ABD)

Now, let's consider triangle BDC. We are given that the measure of angle BDC is twice the measure of angle ABD. Let's denote the measure of angle ABD as x. Therefore, the measure of angle BDC is 2x.

Since the sum of angles in a triangle is 180 degrees, we can write:

x + 2x + angle BCD = 180

3x + angle BCD = 180

angle BCD = 180 - 3x

Now, let's apply the Law of Cosines to triangle BDC to find the length of BC:

BC^2 = BD^2 + CD^2 - 2 * BD * CD * cos(BDC)

BC^2 = 12^2 + 12^2 - 2 * 12 * 12 * cos(2x)

BC^2 = 288 - 288 * cos(2x)

Since AD = DC, we have AD = 12 inches. Now we can write the equation for the total length AC:

AC = AD + DC

AC = 12 + 12

AC = 24 inches

Therefore, the length of AC in triangle ABC is 24 inches.

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Answer:

a) 18.84 cm

b) 28.26 cm²

Step-by-step explanation:

We Know

Each pancake is a circle with a diameter of 6 centimeters.

a)  Calculate the circumference of each pancake.

Circumference of circle = d · π

d = 6 cm

We Take

6 · 3.14 = 18.84 cm

So, the circumference of each pancake is 18.84 cm.

b) Calculate the area of each pancake.

Area of circle = r² · π

r = 1/2 · d

r = 1/2 · 6 = 3 cm

We Take

3² · 3.14 = 28.26 cm²

So, the area of each pancake is 28.26 cm².

The profit is positive, the firm makes a profit of $21,243 at the optimal level of production.

Cost of producing 40 units

We know that the total cost of producing 30 units is $6000. Let's denote the total cost function by C(x), where x is the number of units produced. Then, we have:

C(30) = $6000

The marginal cost function is given as MC = 8x + 70. Integrating this function, we get the total cost function as:

C(x) = [tex]4x^2[/tex] + 70x + C

To find the value of the constant C, we use the fact that C(30) = $6000:

4[tex](30)^2[/tex] + 70(30) + C = $6000

Solving for C, we get:

C = $300

Therefore, the total cost function is:

C(x) = [tex]4x^2[/tex] + 70x + $300

To find the cost of producing 40 units, we evaluate C(40):

C(40) = [tex]4(40)^2[/tex] + 70(40) + $300

C(40) = $7000

Therefore, the cost of producing 40 units is $7000.

Optimal level of production:

The optimal level of production is the value of x that maximizes the profit function. To find this value, we need to set the marginal cost equal to the marginal revenue:

MC = MR

8x + 70 = -6x + 55

Solving for x, we get:

x = 5/7

Since the optimal level of production should be a whole number, we round x up to 1 unit.

Therefore, the optimal level of production is 1 unit.

Profit function:

The profit function is given as:

P(x) = R(x) - C(x)

where R(x) is the revenue function and C(x) is the cost function.

The marginal revenue function is given as MR = -6x + 55. Integrating this function, we get the revenue function as:

R(x) = -[tex]3x^2[/tex] + 55x + D

To find the value of the constant D, we use the fact that the revenue at x = 80 is $14,920:

[tex]-3(80)^2[/tex] + 55(80) + D = $14,920

Solving for D, we get:

D = $21,520

Therefore, the revenue function is:

R(x) = -[tex]3x^2[/tex] + 55x + $21,520

Substituting the cost function and revenue function in the profit function, we get:

P(x) = ([tex]-3x^2[/tex] + 55x + $21,520) - (4x^2 + 25x + $300)

Simplifying, we get:

P(x) = -[tex]7x^2[/tex] + 30x + $21,220

Therefore, the profit function is P(x) = [tex]-7x^2[/tex] + 30x + $21,220.

Profit or loss at the optimal level:

To find the profit or loss at the optimal level, we evaluate the profit function at x = 1:

P(1) = [tex]-7(1)^2[/tex] + 30(1) + $21,220

P(1) = $21,243

Since the profit is positive, the firm makes a profit of $21,243 at the optimal level of production.

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As y varies inversely as x, the value of y when x = 3 is 45.

Inverse proportionality is expressed as:

y ∝ 1/x

Hence:

y = k/x

Where k is the constant of proportionality.

First, we determine the constant of proportionality.

Using the information given in the problem.

When x = 5, y = 27

Substituting these values into the formula, we get:

y = k/x

27 = k/5

k = 135

Now that we have found the value of k, we can use the formula to find y when x = 3. Substituting x = 3 and k = 135, we get:

y = k/x

y = 135/3

y = 45

Therefore, the value of y is 45.

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