https://file.aiquickdraw.com/mathbot/file/1726863546475du4hq4wa.jpg

Based on the image, here are the problems from the text:

- **23. Find the slope of the price function.** (4 points)
- **24. Find the linear price function $ P(x) $.** (4 points)
- **25. Find the simplified Revenue function $ R(x) $.** (4 points)

The problem context says:

- They sell 1 shirt for $9 and a bundle of 3 shirts for $21.

Let's start by answering each question step by step.

---

### 23. Finding the slope of the price function:

We are given:
- The price of 1 shirt is $9.
- The price of 3 shirts is $21.

The slope formula for a linear function is:
\[
m = \frac{{\Delta y}}{{\Delta x}} = \frac{{y_2 - y_1}}{{x_2 - x_1}}
\]
Where $ y_1 $ and $ y_2 $ are the prices, and $ x_1 $ and $ x_2 $ are the quantities sold.

Here:
- $ y_1 = 9 $ (price for 1 shirt, $ x_1 = 1 $),
- $ y_2 = 21 $ (price for 3 shirts, $ x_2 = 3 $).

Thus, the slope $ m $ is:
\[
m = \frac{21 - 9}{3 - 1} = \frac{12}{2} = 6
\]
So, the slope of the price function is **6**.

---

### 24. Finding the linear price function $ P(x) $:

The price function $ P(x) $ will be a linear equation of the form:
\[
P(x) = mx + b
\]
We already know the slope $ m = 6 $.

To find the intercept $ b $, we can use one of the given points. Let's use the point where $ 1 $ shirt costs $9:
\[
P(1) = 6(1) + b = 9
\]
Solving for $ b $:
\[
6 + b = 9 \quad \Rightarrow \quad b = 3
\]

Thus, the linear price function is:
\[
P(x) = 6x + 3
\]

---

### 25. Finding the simplified Revenue function $ R(x) $:

Revenue $ R(x) $ is calculated by multiplying the price function $ P(x) $ by the number of shirts sold $ x $. Therefore:
\[
R(x) = x \cdot P(x)
\]
Substituting the price function $ P(x) = 6x + 3 $:
\[
R(x) = x(6x + 3) = 6x^2 + 3x
\]

Thus, the simplified revenue function is:
\[
R(x) = 6x^2 + 3x
\]

---

Let me know if you'd like further details or have any questions!

### Related Questions:
1. How do you find the slope between two points in general?
2. What is the difference between slope and the y-intercept in a linear equation?
3. How can you derive the price function from more complex sales data?
4. What role does the revenue function play in understanding business profit?
5. How can you find the maximum revenue using the revenue function?

**Tip:** In business math, revenue functions can help determine break-even points or maximum profitability by finding when marginal costs equal marginal revenues.

23. Find the slope of the price function. 24. Find the linear price function P(x). 25. Find the simplified Revenue function R(x). A local animal shelter is selling t-shirts to raise funds for pet rescue efforts. They sell 1 shirt for $9 and a bundle of 3 shirts for $21.

Learn how to calculate the slope of the price function, derive the linear price function P(x), and simplify the revenue function R(x) based on t-shirt sales data from a local shelter. This problem is ideal for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation