Q: If $$x = 82.35$$, what is the weekly cost according to $$CC(x) = 50 + 2.5x$$?

$$CC(82.35) = 50 + 2.5 \times 82.35 = 255.88$$

Q: If $$x = 82.35$$, what is the weekly income according to $$I(x) = 3x + 2 \ln(x)$$?

$$I(82.35) = 3 \times 82.35 + 2 \ln(82.35) \approx 247.05 + 8.36 = 255.41$$

Question 1

Given the weekly cost function $$CC(x) = 50 + 2.5x$$ and the weekly income function $$I(x) = 3x + 2 \ln(x)$$, what is the correct equation for the weekly profit?

Accepted Answer

$$P(x) = 3x + 2 \ln(x) - (50 + 2.5x)$$

Question 2

If the weekly profit function is $$P(x) = 3x + 2 \ln(x) - (50 + 2.5x)$$, what value of $$x$$ maximizes the profit according to the method of successive approximations shown in the notes?

Accepted Answer

$$x \approx 82.35$$

Question 3

What is the derivative of the profit function $$P(x) = 3x + 2 \ln(x) - 50 - 2.5x$$ with respect to $$x$$?

Accepted Answer

$$P'(x) = 3 + \frac{2}{x} - 2.5$$

Question 4

To find the maximum profit, which equation must be solved?

Accepted Answer

$$3 + \frac{2}{x} - 2.5 = 0$$

Question 5

If the initial guess for $$x is$$ $$50$$, what is the next approximation for $$x$$ using the Newton-Raphson method for $$P'(x) = 0$$?

Accepted Answer

$$x_{1} \approx 81.81$$

Question 6

What is the economic interpretation of the solution $$x \approx 82.35 in $$the context of the problem?

Accepted Answer

It is the optimal number of kilograms to produce weekly to maximize profit.

Question 7

Which of the following best describes the method used to solve for the optimal $$x in $$the notes?

Accepted Answer

Successive approximations (Newton-Raphson method)

Question 8

If the cost per kilogram increases from $$2.5 to$$ $$3$$, how would this affect the optimal $$x$$?

Accepted Answer

The optimal $$x$$ would decrease.

Question 9

Suppose the weekly income function changes to $$I(x) = 4x + 2 \ln(x). $$What is the new equation for the profit function?

Accepted Answer

$$P(x) = 4x + 2 \ln(x) - (50 + 2.5x)$$

Question 10

If the fixed weekly cost increases from $$50 to$$ $$60$$, what is the effect on the optimal $$x$$?

Accepted Answer

The optimal $$x$$ remains unchanged.

Question 11

What is the main purpose of using the method of successive approximations in this context?

Accepted Answer

To find the value of $$x$$ that maximizes profit when the equation cannot be solved analytically.

Question 12

If $$x = 82.35$$, what is the weekly cost according to $$CC(x) = 50 + 2.5x$$?

Accepted Answer

$$CC(82.35) = 50 + 2.5 	imes 82.35 = 255.88$$

Question 13

If $$x = 82.35$$, what is the weekly income according to $$I(x) = 3x + 2 \ln(x)$$?

Accepted Answer

$$I(82.35) = 3 	imes 82.35 + 2 \ln(82.35) \approx 247.05 + 8.36 = 255.41$$

Question 14

What is the weekly profit when $$x = 82.35$$?

Accepted Answer

$$P(82.35) = 255.41 - 255.88 = -0.47$$

Question 15

If the weekly profit is negative at the optimal $$x$$, what should the business consider?

Accepted Answer

Re-evaluate costs and pricing to ensure profitability.

Question 16

Which of the following is NOT a plausible reason for using logarithmic terms in the income function?

Accepted Answer

To represent fixed costs.

Question 17

If the cost function is $$CC(x) = 50 + 2.5x$$ and the income function is $$I(x) = 3x + 2 \ln(x)$$, what is the break-even point?

Accepted Answer

The value of $$x$$ where $$I(x) = CC(x)$$

Optimization Methods: Secant Method for Maximizing Weekly Profit

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