https://file.aiquickdraw.com/mathbot/file/1726681500963zai5nllg.jpg

This problem deals with the appreciation in the value of a painting. Let's break it down and solve it.

### Given Information:
- Initial value of the painting: \$50,000
- The painting appreciates at a constant rate of \$5,000 per year.
- We are asked to predict the value of the painting after 3 years.

### Approach:

We can use the equation of a line (linear equation) to represent the value of the painting over time, since the value increases at a constant rate. This is a case of simple linear appreciation.

The general form of the linear equation is:
\[
V(t) = V_0 + r \cdot t
\]
Where:
- $V(t)$ is the value of the painting at time $t$,
- $V_0$ is the initial value (at time $t = 0$),
- $r$ is the constant rate of appreciation (in this case, \$5,000 per year),
- $t$ is the number of years since the purchase.

### Step-by-step Solution:

1. **Write the equation**: 
   - The initial value $V_0 = 50,000$,
   - The rate of appreciation $r = 5,000$.
   
   So, the equation becomes:
   \[
   V(t) = 50,000 + 5,000 \cdot t
   \]

2. **Calculate the value after 3 years**:
   To find the value of the painting after 3 years, substitute $t = 3$ into the equation:
   \[
   V(3) = 50,000 + 5,000 \cdot 3
   \]
   \[
   V(3) = 50,000 + 15,000 = 65,000
   \]

### Final Answer:
The value of the painting after 3 years will be **\$65,000**.

Would you like further details on this problem, or clarification on any steps?

---

Here are 5 follow-up questions to expand on this concept:
1. What will be the value of the painting after 5 years?
2. How would the equation change if the rate of appreciation were \$7,000 per year instead?
3. What if the painting depreciated at \$5,000 per year instead of appreciating? What would the value be after 3 years?
4. How long will it take for the painting to reach a value of \$80,000?
5. Can you express the equation of the value as a function of time $t$ in years, where the rate of appreciation is variable?

**Tip**: In problems involving constant rates (appreciation or depreciation), a linear model is often used. Make sure to always check if the rate is constant to apply this model properly.

Suppose a painting purchased for $50,000 is expected to appreciate in value at a constant rate of $5,000 per year for the next 5 years. Use Formula (5) to write an equation predicting the value of the painting in the next several years. What will be its value 3 years from the date of purchase?

Learn how to calculate the value of a painting that appreciates at a constant rate of $5,000 per year, starting from an initial value of $50,000. This example demonstrates how to predict the value after 3 years using a simple linear equation. Suitable for 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