https://file.aiquickdraw.com/mathbot/file/1725487154674oasc77bz.jpg

From the image you provided, the problem seems to involve a linear function, $ P(t) $, that models the price of a used car based on its age, $ t $, in years. Specifically, it is known that:

\[
P(1) = 7300 \quad \text{and} \quad P(7) = 5300.
\]

You are asked to determine which statement best describes how the price decreases per year.

### Solution:

Since this is a linear function, the price decreases by the same amount each year, which means the slope $ m $ of the function $ P(t) $ is constant.

To find the slope $ m $, we can use the slope formula for a linear function:

\[
m = \frac{P(7) - P(1)}{7 - 1} = \frac{5300 - 7300}{7 - 1} = \frac{-2000}{6} = -333.33
\]

This means the price of the car decreases by approximately $333.33 per year.

The correct answer in the image corresponds to the statement:
"For each year that a car ages, its price decreases by approximately $333."

### Would you like further details on how linear models work or have any questions?

### Related Questions:
1. How would the linear model change if the car's price increases with age?
2. How do you interpret the slope in other real-life contexts?
3. Can a linear function ever model a car's value increasing over time?
4. What happens to the function $ P(t) $ if we model a faster depreciation rate?
5. How would this problem change if you had a different type of function, like quadratic depreciation?

### Tip:
Linear functions are a great way to model constant rates of change, but always check if your situation involves constant or changing rates!

A linear function P(t) used to model the price, in dollars, of used cars as a function of their age in years. It is known that P(1) = 7300 and P(7) = 5300. Based on this model, which of the following is true?

Explore how a linear function models the depreciation of a used car's price based on its age. This problem involves calculating the constant rate of decrease using the slope formula. Suitable for Grades 9-12.

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