https://file.aiquickdraw.com/mathbot/file/17266182700181fxd6822.jpg

The problem in the image asks us to solve the following:

A company manufactures cardboard boxes. The fixed costs incurred by the company are **5,280 soles**, and the cost of producing one box is **1.15 soles**. The selling price for each box is **3.55 soles**.

### a) Represent the profit function (Función utilidad):
Let \( x \) be the number of boxes sold.

The total revenue \( R(x) \) is given by the selling price per box multiplied by the number of boxes sold:
\[
R(x) = 3.55x
\]

The total cost \( C(x) \) is the sum of the fixed costs and the variable costs (cost per box multiplied by the number of boxes produced):
\[
C(x) = 5,280 + 1.15x
\]

The profit \( P(x) \) is the difference between revenue and total cost:
\[
P(x) = R(x) - C(x) = 3.55x - (5,280 + 1.15x)
\]
Simplifying this:
\[
P(x) = 3.55x - 5,280 - 1.15x = 2.40x - 5,280
\]

Thus, the profit function is:
\[
P(x) = 2.40x - 5,280
\]

### b) How many units must be sold for the profit to exceed 2,700 soles?
We want the profit to be greater than 2,700 soles, so we solve the inequality:
\[
2.40x - 5,280 > 2,700
\]

Let's solve for \( x \):

\[
2.40x > 2,700 + 5,280
\]
\[
2.40x > 7,980
\]
\[
x > \frac{7,980}{2.40}
\]
\[
x > 3,325
\]

Therefore, the company must sell **at least 3,326 boxes** to exceed a profit of 2,700 soles.

---

Would you like more details on the steps or further clarification? Here are 5 related questions to explore:

1. How would the profit function change if the cost per box increased?
2. What would be the breakeven point, where profit equals zero?
3. How does the fixed cost impact the number of units that need to be sold for profitability?
4. How would a change in the selling price affect the profit function?
5. What happens to the profit function if the fixed costs decrease by 10%?

**Tip**: When calculating profit functions, always start by breaking down revenue and total costs separately before combining them.

Una empresa manufacturera elabora cajas de cartón. Los costos fijos en los que incurre la empresa ascienden a 5,280 soles y los costos de producir una caja es 1.15 soles. Si el precio de venta de cada caja es de 3.55 soles. a) Represente la función utilidad. b) Cuántas unidades debe vender para que la utilidad supere los 2,700 soles.

This problem helps you calculate the profit function for a manufacturing company that produces cardboard boxes. Learn how to find the number of units needed to exceed a profit of 2,700 soles based on fixed and variable costs.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation