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 profit, which equation should you solve based on the given functions?

Accepted Answer

$$\frac{d}{dx} [3x + 2 \ln(x) - (50 + 2.5x)] = 0$$

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 set the derivative of the profit function to zero, which equation do you get?

Accepted Answer

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

Question 5

Solve 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

If you start with $$x_0 = 50$ in the Newton-Raphson method, what is the value of f$$($$x_0$$)$$ for f(x) = 0.5$$ + \frac{2}{x}$$?

Accepted Answer

$$0.54$$

Question 9

What is the economic interpretation of the point where the profit function's derivative equals zero?

Accepted Answer

It is the production level that maximizes profit.

Question 10

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

Accepted Answer

The profit-maximizing quantity does not change.

Question 11

Suppose the price per kilogram increases from $$3 to$$ $$4. $$How does this affect the profit-maximizing quantity?

Accepted Answer

The profit-maximizing quantity increases.

Question 12

If the cost per kilogram increases from $$2.5 to$$ $$3$$, what is the new equation to solve for the profit-maximizing quantity?

Accepted Answer

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

Optimization Methods: Solving a Production Cost Problem Using Iterative Techniques

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