Q: If the weekly profit function is $$P(x) = 3x + 2 \ln(x) - 50 - 2.5x$$, what is the profit for $$x = 10$$?

$$3 \times 10 + 2 \ln(10) - 50 - 2.5 \times 10$$

Question 1

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

Accepted Answer

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

Question 2

If the weekly profit function is $$P(x) = 3x + 2 \ln(x) - 50 - 2.5x$$, what value of $$x$$ maximizes the profit?

Accepted Answer

The value of $$x$$ that solves $$3 + \frac{2}{x} - 2.5 = 0$$

Question 3

What is the economic interpretation of the equation $$3x + 2 \ln(x) = 50 + 2.5x$$?

Accepted Answer

It represents the break-even point where income equals cost.

Question 4

If $$x is $$the number of kilograms sold per week, what does the term $$2 \ln(x) in $$the income function represent?

Accepted Answer

Additional income due to logarithmic growth in sales.

Question 5

Suppose the cost function is $$CC(x) = 50 + 2.5x. $$What is the marginal cost for producing one more kilogram?

Accepted Answer

$$2.5$$

Question 6

If the weekly income is $$I(x) = 3x + 2 \ln(x)$$, what is the marginal income for one more kilogram sold?

Accepted Answer

$$3 + \frac{2}{x}$$

Question 7

A business wants to maximize its weekly profit. Which equation should it solve?

Accepted Answer

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

Question 8

If the initial guess for $$x is$$ $$50$$, and the Newton-Raphson method is used to solve $$3 + \frac{2}{x} - 2.5 = 0$$, what is the formula for the next iteration?

Accepted Answer

$$x_{k+1} = x_k - \frac{3 + \frac{2}{x_k} - 2.5}{-\frac{2}{x_k^2$}}$$

Question 9

If the cost per kilogram increases to $$3$$ instead of $$2.5$$, how does the equation for maximizing profit change?

Accepted Answer

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

Question 10

If the fixed weekly cost increases from $$50 to$$ $$60$$, what happens to the break-even point?

Accepted Answer

The break-even point increases; more kilograms must be sold to cover costs.

Question 11

If the weekly income function is $$I(x) = 3x + 2 \ln(x)$$, what is the income for selling $$10$$ kilograms?

Accepted Answer

$$3 	imes 10 + 2 \ln(10)$$

Question 12

If the cost function is $$CC(x) = 50 + 2.5x$$, what is the total cost for producing $$20$$ kilograms?

Accepted Answer

$$50 + 2.5 	imes 20$$

Question 13

If the weekly profit function is $$P(x) = 3x + 2 \ln(x) - 50 - 2.5x$$, what is the profit for $$x = 10$$?

Accepted Answer

$$3 	imes 10 + 2 \ln(10) - 50 - 2.5 	imes 10$$

Question 14

If the weekly cost function is $$CC(x) = 50 + 2.5x$$, what is the fixed cost?

Accepted Answer

$$50$$

Question 15

If the weekly income function is $$I(x) = 3x + 2 \ln(x)$$, what is the variable income per kilogram?

Accepted Answer

$$3$$

Question 16

If the cost function is $$CC(x) = 50 + 2.5x$$ and the income function is $$I(x) = 3x + 2 \ln(x)$$, what is the minimum number of kilograms needed to break even?

Accepted Answer

The value of $$x$$ that solves $$3x + 2 \ln(x) = 50 + 2.5x$$

Question 17

If the weekly profit function is $$P(x) = 3x + 2 \ln(x) - 50 - 2.5x$$, what is the derivative with respect to $$x$$?

Accepted Answer

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

Optimization Using the Secant Method: Cost and Revenue Analysis

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