Q: What is the weekly profit if $$x = 8$$ kilograms are sold, using the original cost and income functions?

$$P(8) = 3 \times 8 + 2 \ln(8) - (50 + 2.5 \times 8)$$

Question 1

Given the weekly cost function for producing a chemical:

$$CC(x) = 50 + 2.5x$$

and the weekly income function:

$$R(x) = 3x + 2 \ln(x)$$

where $$x is $$the number of kilograms sold per week, 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, what equation must be solved for $$x$$?

Accepted Answer

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

Question 3

What is the economic interpretation of the equation $$3 + \frac{2}{x} = 2.5 in $$this context?

Accepted Answer

The marginal income equals the marginal cost.

Question 4

If $$x = 8.18 is an $$approximate solution to the profit maximization equation, what does this value represent?

Accepted Answer

The number of kilograms to sell weekly to maximize profit.

Question 5

Which method is being used to solve the equation $$3 + \frac{2}{x} - 2.5 = 0$$ iteratively in the notes?

Accepted Answer

The secant method.

Question 6

What is the formula for the secant method as used in the notes?

Accepted Answer

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

Question 7

Suppose the cost per kilogram increases to $$3$$ instead of $$2.5. $$What is the new equation to maximize profit?

Accepted Answer

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

Question 8

If the fixed weekly cost increases to $$60$$, how does this affect the value of $$x$$ that maximizes profit?

Accepted Answer

It does not affect the value of $$x$$ that maximizes profit.

Question 9

What is the weekly profit if $$x = 8$$ kilograms are sold, using the original cost and income functions?

Accepted Answer

$$P(8) = 3 	imes 8 + 2 \ln(8) - (50 + 2.5 	imes 8)$$

Question 10

In a real-world scenario, why might the income function include a logarithmic term $$2 \ln(x)$$?

Accepted Answer

To model diminishing returns or market saturation effects.

Question 11

If the weekly cost function is $$CC(x) = 50 + 2.5x$$ and the weekly income is $$R(x) = 3x + 2 \ln(x)$$, what is the break-even point (where profit is zero)?

Accepted Answer

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

Question 12

If the price per kilogram increases, what happens to the optimal value of $$x$$ that maximizes profit?

Accepted Answer

The optimal value of $$x$$ increases.

Question 13

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

Accepted Answer

It is used to numerically solve the nonlinear equation for profit maximization.

Question 14

If the weekly cost function becomes $$CC(x) = 50 + 2.5x + 0.1x^2$, how does this affect the profit maximization equation?

Accepted Answer

$$0 = 3 + \frac{2}{x} - 2.5 - 0.2x$$

Método de la Secante: Aplicación en Optimización de Costos

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