A firm produces a product that has the production cost function C(x) = 440x + 6270 and the revenue function R(x) = 550x. Find the break-even quantity, then find the profit function. units. The break-even quantity is (Type a whole number.) Write the profit function. P(x) = 0
A firm produces a product that has the production cost function C(x) = 440x + 6270 and the revenue function R(x) = 550x. Find the break-even quantity, then find the profit function. units. The break-even quantity is (Type a whole number.) Write the profit function. P(x) = 0
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
12
![**Understanding Break-Even Quantity and Profit Function**
A firm produces a product with the following production cost and revenue functions:
- Production Cost Function: \( C(x) = 440x + 6270 \)
- Revenue Function: \( R(x) = 550x \)
To find the break-even quantity and the profit function:
1. **Break-Even Quantity:**
The break-even point occurs when the total revenue equals the total cost. Mathematically, this is where \( R(x) = C(x) \).
Given:
\[
R(x) = 550x
\]
\[
C(x) = 440x + 6270
\]
Set the two equations equal to each other to solve for \( x \):
\[
550x = 440x + 6270
\]
Simplify the equation:
\[
550x - 440x = 6270
\]
\[
110x = 6270
\]
\[
x = \frac{6270}{110}
\]
\[
x = 57
\]
Therefore, the break-even quantity is **57 units**.
2. **Profit Function:**
The profit \( P(x) \) is given by the revenue minus the cost:
\[
P(x) = R(x) - C(x)
\]
Substituting the given functions:
\[
P(x) = 550x - (440x + 6270)
\]
\[
P(x) = 550x - 440x - 6270
\]
\[
P(x) = 110x - 6270
\]
Therefore, the profit function is \( P(x) = 110x - 6270 \).
**Key Points:**
- The break-even quantity is the number of units that must be produced and sold for total revenue to equal total cost.
- The profit function represents the firm's profit depending on the number of units produced and sold.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F11a64bba-74c0-4fd7-b16b-3d8fff0a6ac2%2F346e1f37-5f39-45da-976d-588a27a61a17%2Fsbygkbp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Understanding Break-Even Quantity and Profit Function**
A firm produces a product with the following production cost and revenue functions:
- Production Cost Function: \( C(x) = 440x + 6270 \)
- Revenue Function: \( R(x) = 550x \)
To find the break-even quantity and the profit function:
1. **Break-Even Quantity:**
The break-even point occurs when the total revenue equals the total cost. Mathematically, this is where \( R(x) = C(x) \).
Given:
\[
R(x) = 550x
\]
\[
C(x) = 440x + 6270
\]
Set the two equations equal to each other to solve for \( x \):
\[
550x = 440x + 6270
\]
Simplify the equation:
\[
550x - 440x = 6270
\]
\[
110x = 6270
\]
\[
x = \frac{6270}{110}
\]
\[
x = 57
\]
Therefore, the break-even quantity is **57 units**.
2. **Profit Function:**
The profit \( P(x) \) is given by the revenue minus the cost:
\[
P(x) = R(x) - C(x)
\]
Substituting the given functions:
\[
P(x) = 550x - (440x + 6270)
\]
\[
P(x) = 550x - 440x - 6270
\]
\[
P(x) = 110x - 6270
\]
Therefore, the profit function is \( P(x) = 110x - 6270 \).
**Key Points:**
- The break-even quantity is the number of units that must be produced and sold for total revenue to equal total cost.
- The profit function represents the firm's profit depending on the number of units produced and sold.
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