Q: If the cost function is $$CC(x) = 50 + 2.5x$$ and the income function is $$I(x) = 3x + 2 \ln(x)$$, what is the profit when $$x = 10$$?

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

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

The weekly income from selling $$x kg of a $$chemical is given by $$I(x) = 3x + 2 \ln(x). $$What is the income from selling 5 kg?

Accepted Answer

$$I(5) = 15 + 2 \ln(5)$$

Question 3

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

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 kg increases to $$3$$, what is the new cost function?

Accepted Answer

$$CC(x) = 50 + 3x$$

Question 7

Suppose the weekly income function is $$I(x) = 3x + 2 \ln(x)$$ and the cost function is $$CC(x) = 50 + 2.5x. $$What is the break-even point?

Accepted Answer

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

Question 8

If the weekly cost is $$CC(x) = 50 + 2.5x$$ and you want to keep costs below $$100$$, what is the maximum $$x$$ you can produce?

Accepted Answer

$$x \u003c 25$$

Question 9

If the weekly income is $$I(x) = 3x + 2 \ln(x)$$, which of the following best describes the effect of increasing $$x$$?

Accepted Answer

Income increases, but at a decreasing rate due to the logarithmic term.

Question 10

In the context of the Newton-Raphson method, which equation is used to find the root for maximizing profit?

Accepted Answer

$$x_{k+1} = x_k - \frac{f(x_k)}{f'(x_k)}$$

Question 11

If you start with $$x_0 = 50$ and apply the Newton-Raphson method to f(x) = 3$$ + \frac{2}{x} - 2.5$$, what is the next approximation x_1$?

Accepted Answer

$$x_1 = 81.81$

Question 12

What is the main advantage of using the Newton-Raphson method in this scenario?

Accepted Answer

It quickly finds the value of $$x$$ that maximizes profit.

Question 13

If the cost function is $$CC(x) = 50 + 2.5x$$ and the income function is $$I(x) = 3x + 2 \ln(x)$$, what is the profit when $$x = 10$$?

Accepted Answer

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

Question 14

If the weekly cost increases by a fixed amount each week, which part of the cost function would change?

Accepted Answer

The constant term (e.g., $$50 in$$ $$CC(x) = 50 + 2.5x)$$

Question 15

If the price per kg sold increases, which part of the income function $$I(x) = 3x + 2 \ln(x)$$ would change?

Accepted Answer

The coefficient of $$x$$

Optimization of Weekly Production Using Algebraic Modeling

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