### Retail Profit Problem**Problem Statement:**It costs Liz $12 to build a picture frame. She estimates that, if she charges $ x $ dollars per frame, she can sell $ 60 - x $ frames per week. Write a function for her weekly profit.**Solution:**To determine Liz's weekly profit, we need to account for her revenue and her costs. The revenue is the amount of money she receives from selling the frames, while the costs are the expenses she incurs to produce those frames.1. **Cost per Frame:** Liz's cost to build a picture frame is $12.2. **Selling Price per Frame:** Let $ x $ be the price Liz charges per frame.3. **Number of Frames Sold per Week:** According to her estimation, she can sell $ 60 - x $ frames per week if the price per frame is $ x $.4. **Total Revenue per Week:** - Revenue is calculated by multiplying the price per frame $ x $ by the number of frames sold $ 60 - x $. - Therefore, the total revenue per week is: \[ \text{Revenue} = x \times (60 - x) = 60x - x^2 \]5. **Total Cost per Week:** - Cost is calculated by multiplying the cost per frame ($12) by the number of frames sold $ 60 - x $. - Therefore, the total cost per week is: \[ \text{Cost} = 12 \times (60 - x) = 720 - 12x \]6. **Weekly Profit Function:** - Profit is the difference between total revenue and total cost. - Therefore, the profit $ P(x) $ can be expressed as: \[ P(x) = \text{Revenue} - \text{Cost} \] \[ P(x) = (60x - x^2) - (720 - 12x) \] \[ P(x) = 60x - x^2 - 720 + 12x \] \[ P(x) = -x^2 + 72x - 720 \]So, the function for Liz's weekly profit is:\[ P(x) = -x^2 + 72x - 720 \]

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Retail profit. It costs Liz $12 to build a picture frame. She estimates that, if she charges x dollars per frame, she can sell 60 – x frames per week. Write a function for her weekly profit.

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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### Retail Profit Problem

**Problem Statement:**

It costs Liz $12 to build a picture frame. She estimates that, if she charges $ x $ dollars per frame, she can sell $ 60 - x $ frames per week. Write a function for her weekly profit.

**Solution:**

To determine Liz's weekly profit, we need to account for her revenue and her costs. The revenue is the amount of money she receives from selling the frames, while the costs are the expenses she incurs to produce those frames.

1. **Cost per Frame:** Liz's cost to build a picture frame is $12.

2. **Selling Price per Frame:** Let $ x $ be the price Liz charges per frame.

3. **Number of Frames Sold per Week:** According to her estimation, she can sell $ 60 - x $ frames per week if the price per frame is $ x $.

4. **Total Revenue per Week:**
- Revenue is calculated by multiplying the price per frame $ x $ by the number of frames sold $ 60 - x $.
- Therefore, the total revenue per week is:
\[ \text{Revenue} = x \times (60 - x) = 60x - x^2 \]

5. **Total Cost per Week:**
- Cost is calculated by multiplying the cost per frame ($12) by the number of frames sold $ 60 - x $.
- Therefore, the total cost per week is:
\[ \text{Cost} = 12 \times (60 - x) = 720 - 12x \]

6. **Weekly Profit Function:**
- Profit is the difference between total revenue and total cost.
- Therefore, the profit $ P(x) $ can be expressed as:
\[ P(x) = \text{Revenue} - \text{Cost} \]
\[ P(x) = (60x - x^2) - (720 - 12x) \]
\[ P(x) = 60x - x^2 - 720 + 12x \]
\[ P(x) = -x^2 + 72x - 720 \]

So, the function for Liz's weekly profit is:
\[ P(x) = -x^2 + 72x - 720 \]

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