https://file.aiquickdraw.com/mathbot/file/1729198924175gvcnh6h9.jpg

The problem provides two functions:
1. **Cost function:** \( C(q) = 80 + 13q \)
2. **Price function:** \( p = 73 - 2q \)

We are tasked with finding:
a) The quantity \( q \) that produces maximum profit,
b) The price \( p \) per unit at this \( q \),
c) The maximum profit \( P \).

### Steps to solve:

1. **Revenue function** \( R(q) \):
   Revenue is the product of price and quantity:
   \[
   R(q) = p \cdot q = (73 - 2q) \cdot q = 73q - 2q^2
   \]

2. **Profit function** \( P(q) \):
   Profit is revenue minus cost:
   \[
   P(q) = R(q) - C(q) = (73q - 2q^2) - (80 + 13q) = 73q - 2q^2 - 80 - 13q
   \]
   Simplifying the profit function:
   \[
   P(q) = -2q^2 + 60q - 80
   \]

3. **Maximizing profit**:
   To find the maximum profit, take the derivative of the profit function and set it equal to zero:
   \[
   \frac{dP}{dq} = -4q + 60
   \]
   Setting the derivative to zero:
   \[
   -4q + 60 = 0
   \]
   Solving for \( q \):
   \[
   q = \frac{60}{4} = 15
   \]

Thus, the number of units that produces maximum profit is **\( q = 15 \)**.

---

Would you like the detailed steps for the next parts? Here are related questions:

1. How do you calculate the price per unit at the maximum profit?
2. What are the steps for determining the maximum profit?
3. What is the significance of the second derivative in confirming a maximum?
4. How can this process be applied to other cost-revenue scenarios?
5. Why does setting the derivative to zero give the maximum?

**Tip**: Maximizing profit often involves solving a quadratic equation derived from the profit function.

For the cost and price functions below, find a) the number, q, of units that produces maximum profit; b) the price, p, per unit that produces maximum profit; and c) the maximum profit, P.

C(q) = 80 + 13q; p = 73 - 2q

Learn how to maximize profit by using cost and price functions with step-by-step optimization techniques. This problem involves setting up revenue and profit functions, applying derivatives, and solving quadratic equations. Suitable for students in Grades 11-12 and early college-level courses.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation