Question 1

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

Accepted Answer

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

Question 2

If you want to maximize the weekly profit $$P(x)$$, which equation should you solve?

Accepted Answer

$$\frac{dP}{dx} = 0$$

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

$$\frac{dP}{dx} = 0.5 + \frac{2}{x}$$

Question 4

To find the optimal $$x$$ that maximizes profit, you set $$\frac{dP}{dx} = 0. $$What equation do you solve?

Accepted Answer

$$0.5 + \frac{2}{x} = 0$$

Question 5

What is the solution for $$x in $$the equation $$0.5 + \frac{2}{x} = 0$$?

Accepted Answer

$$x = -4$$

Question 6

Why is the solution $$x = -4$$ not valid in the context of this problem?

Accepted Answer

Because the number of kilograms sold cannot be negative.

Question 7

If the Newton-Raphson method is used to solve $$0.5 + \frac{2}{x} = 0$$, what is the iterative formula?

Accepted Answer

$$x_{k+1} = x_k - \frac{0.5 + \frac{2}{x_k}}{-\frac{2}{x_k^2$}}$$

Question 8

Suppose you start with $$x_0 = 50$ in the Newton-Raphson method. What is the value of x_1$ after one iteration?

Accepted Answer

$$x_1$$ \approx 81.81$$

Question 9

What is the economic interpretation of the point where $$\frac{dP}{dx} = 0$$?

Accepted Answer

It is the quantity of product that maximizes weekly profit.

Question 10

If the weekly cost increases to $$CC(x) = 60 + 2.5x$$, how does this affect the optimal $$x$$?

Accepted Answer

The optimal $$x$$ does not change.

Question 11

If the price per kilogram increases from 3 to 4, what is the new profit function?

Accepted Answer

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

Question 12

Which of the following best describes the role of the logarithmic term $$2 \ln(x) in $$the income function?

Accepted Answer

It models diminishing returns as more product is sold.

Question 13

If the cost per kilogram increases, what happens to the profit-maximizing quantity?

Accepted Answer

It decreases.

Question 14

Suppose the weekly fixed cost increases. What is the effect on the profit-maximizing quantity?

Accepted Answer

There is no effect.

Question 15

If $$x$$ must be an integer, what is the best way to choose the optimal $$x$$ after solving the continuous problem?

Accepted Answer

Round to the nearest integer.

Optimization Methods: Solving a Cost-Income Model Using Iterative Techniques

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