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 profit function $$P(x)$$?

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

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

Question 4

If you use the secant method to solve $$0.5 + \frac{2}{x} = 0$$, what is the next approximation $$x_{k+1}$$ given $$x_0 = 50$ and x_1$ = 81.81$$?

Accepted Answer

$$x_{2} = 82.35$$

Question 5

What real-world scenario does this problem model?

Accepted Answer

Maximizing the profit for the weekly sale of a chemical product

Question 6

Which of the following best describes the secant method used in this context?

Accepted Answer

An iterative numerical method to approximate roots of a nonlinear equation

Question 7

If the weekly cost increases to $$CC(x) = 50 + 3x$$, how does the equation for maximizing profit change?

Accepted Answer

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

Question 8

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

Accepted Answer

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

Question 9

What is the economic interpretation of the solution to $$0.5 + \frac{2}{x} = 0$$?

Accepted Answer

The number of kilograms to sell weekly to maximize profit

Question 10

If the cost per kilogram increases, what happens to the optimal number of kilograms to sell to maximize profit?

Accepted Answer

It decreases

Question 11

Which of the following is a necessary condition for a maximum in the profit function $$P(x)$$?

Accepted Answer

$$\frac{d^2$}{dx^2$}P(x) \u003c 0$$

Question 12

If $$x$$ must be an integer, what is the best approach to find the optimal $$x$$ after using the secant method?

Accepted Answer

Round the result to the nearest integer and check which integer gives the highest profit

Optimization Using the Secant Method: Cost and Revenue Functions

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