Question 1

Given the weekly cost function for producing a chemical is $$CC(x) = 50 + 2.5x$$, where $$x is $$the number of kilograms produced, what is the cost of producing 10 kg in a week?

Accepted Answer

$$75$$

Question 2

If the weekly income from selling $$x kg of $$the chemical is $$I(x) = 3x + 2 \ln(x)$$, what is the income from selling 5 kg?

Accepted Answer

$$18.22$$

Question 3

What is the profit function $$P(x)$$, given the cost function $$CC(x) = 50 + 2.5x$$ and the income function $$I(x) = 3x + 2 \ln(x)$$?

Accepted Answer

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

Question 4

To maximize profit, you set the derivative of the profit function to zero. What equation do you solve?

Accepted Answer

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

Question 5

What is the critical value of $$x$$ that maximizes profit, based on the equation $$3 + \frac{2}{x} - 2.5 = 0$$?

Accepted Answer

$$x = 4$$

Question 6

If the cost per kilogram increases to $$3$$, making the cost function $$CC(x) = 50 + 3x$$, how does this affect the profit-maximizing quantity?

Accepted Answer

The profit-maximizing quantity decreases.

Question 7

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

Accepted Answer

$$4 + \frac{2}{x} - 2.5 = 0$$

Question 8

If the fixed cost increases to $$100$$, what is the effect on the profit-maximizing quantity?

Accepted Answer

The profit-maximizing quantity does not change.

Question 9

Which of the following best describes the role of the natural logarithm in the income function $$I(x) = 3x + 2 \ln(x)$$?

Accepted Answer

It models diminishing returns as $$x$$ increases.

Question 10

If you use the secant method to solve $$3 + \frac{2}{x} - 2.5 = 0$$, and your first two guesses are $$x_0 = 50$ and x_1$ = 81.81$$, what is the next approximation $$x_2$$?

Accepted Answer

$$x_2 = 82.35$

Question 11

Why is it important to use iterative methods like the secant method for equations such as $$3 + \frac{2}{x} - 2.5 = 0$$?

Accepted Answer

Because the equation cannot be solved algebraically for $$x$$ easily.

Question 12

If the weekly cost function is $$CC(x) = 50 + 2.5x$$ and the weekly income is $$I(x) = 3x + 2 \ln(x)$$, what is the profit when $$x = 4$$?

Accepted Answer

$$P(4) = 12 + 2 \ln(4) - 50 - 10 = -46.23$$

Question 13

If the cost per kilogram is reduced to $$2$$, what is the new profit-maximizing equation?

Accepted Answer

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

Question 14

Which of the following is a real-world reason for including a logarithmic term in the income function?

Accepted Answer

To model a situation where each additional unit sold brings less additional income.

Optimization of Weekly Production Using Algebraic Methods

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