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 $$P(x)$$?

Accepted Answer

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

Question 2

To maximize the weekly profit, which equation must be solved for $$x$$?

Accepted Answer

$$\frac{d}{dx}P(x) = 0$$

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

If you set $$\frac{dP}{dx} = 0$$ for $$P(x) = 3x + 2 \ln(x) - 50 - 2.5x$$, what equation do you get?

Accepted Answer

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

Question 5

Solve $$0.5 + \frac{2}{x} = 0$$ for $$x.$$

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

If the initial guess for $$x is$$ $$50$$, what is the value of the function $$f(x) = 0.5 + \frac{2}{x} at $$this point?

Accepted Answer

$$0.54$$

Question 9

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

Accepted Answer

It is the value of $$x$$ that maximizes the weekly profit.

Question 10

Suppose the cost per kilogram increases to $$3$$ instead of $$2.5. $$How does this affect the profit-maximizing quantity?

Accepted Answer

The profit-maximizing quantity decreases.

Question 11

If the weekly fixed cost increases from $$50 to$$ $$100$$, what happens to the profit-maximizing quantity?

Accepted Answer

It remains unchanged.

Question 12

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

Accepted Answer

It represents diminishing returns to scale as $$x$$ increases.

Question 13

If $$x = 82.35 is $$found as a solution using the Newton-Raphson method, what does this value represent?

Accepted Answer

The approximate number of kilograms to sell weekly to maximize profit.

Question 14

If the weekly income is less than the weekly cost for all $$x \u003e 0$$, what can be concluded?

Accepted Answer

There is no value of $$x$$ that yields a profit.

Optimization Methods: Solving for Maximum Profit in a Weekly Production Model

Guida di studio - Domande di pratica

Study guide - Flashcard