Question
Raven gets a brand new Honda Civic for her high school graduation! The car costs $24,250. The car loses 15% of its value each year.
1. Write a function to model the value of the car over time.
2. What will be the value of the car after 10 years? Round to the nearest cent.
3. After how many years will the car be worth $20,000? Round your answer to the nearest tenth. Find the answer algebraically.
You must show thorough, handwritten work. You may only use scratch paper, a writing utensil, and a calculator that does not have a computer algebra system.
1. Write a function to model the value of the car over time.
2. What will be the value of the car after 10 years? Round to the nearest cent.
3. After how many years will the car be worth $20,000? Round your answer to the nearest tenth. Find the answer algebraically.
You must show thorough, handwritten work. You may only use scratch paper, a writing utensil, and a calculator that does not have a computer algebra system.
Answers
GPT-4o mini
Let’s break down the problem step by step.
### 1. Write a function to model the value of the car over time.
The value of the car decreases by 15% each year. This means that after one year, the car's value will be 85% of its original value (100% - 15% = 85%).
Let \( V(t) \) represent the value of the car after \( t \) years. We can express this in mathematical terms as follows:
- The initial value of the car \( V(0) = 24,250 \).
The value of the car after \( t \) years can be modeled by the equation:
\[
V(t) = V(0) \times (1 - r)^t
\]
where \( r \) is the rate of depreciation (0.15 for 15%).
Substituting into the equation:
\[
V(t) = 24,250 \times (0.85)^t
\]
### 2. What will be the value of the car after 10 years?
To find \( V(10) \):
\[
V(10) = 24,250 \times (0.85)^{10}
\]
Calculating \( (0.85)^{10} \):
Using a calculator,
\[
(0.85)^{10} \approx 0.196874404
\]
Now, substituting this value back into the equation:
\[
V(10) = 24,250 \times 0.196874404 \approx 4,772.50
\]
Thus, the value of the car after 10 years is approximately
\[
\boxed{4,772.50}
\]
### 3. After how many years will the car be worth $20,000?
We want to find \( t \) such that \( V(t) = 20,000 \).
Setting up the equation:
\[
20,000 = 24,250 \times (0.85)^t
\]
First, divide both sides by 24,250:
\[
\frac{20,000}{24,250} = (0.85)^t
\]
Calculating \( \frac{20,000}{24,250} \):
\[
\frac{20,000}{24,250} \approx 0.8251
\]
Now, we have:
\[
0.8251 \approx (0.85)^t
\]
To solve for \( t \), we take the logarithm of both sides:
\[
\log(0.8251) = \log((0.85)^t)
\]
Using the power rule of logarithms:
\[
\log(0.8251) = t \cdot \log(0.85)
\]
Now solve for \( t \):
\[
t = \frac{\log(0.8251)}{\log(0.85)}
\]
Calculating the logarithms using a calculator:
\[
\log(0.8251) \approx -0.0822 \quad \text{and} \quad \log(0.85) \approx -0.0706
\]
Thus,
\[
t = \frac{-0.0822}{-0.0706} \approx 1.164
\]
Finally, rounding to the nearest tenth:
\[
t \approx 1.2
\]
So, the car will be worth $20,000 after about
\[
\boxed{1.2} \text{ years}.
\]
### 1. Write a function to model the value of the car over time.
The value of the car decreases by 15% each year. This means that after one year, the car's value will be 85% of its original value (100% - 15% = 85%).
Let \( V(t) \) represent the value of the car after \( t \) years. We can express this in mathematical terms as follows:
- The initial value of the car \( V(0) = 24,250 \).
The value of the car after \( t \) years can be modeled by the equation:
\[
V(t) = V(0) \times (1 - r)^t
\]
where \( r \) is the rate of depreciation (0.15 for 15%).
Substituting into the equation:
\[
V(t) = 24,250 \times (0.85)^t
\]
### 2. What will be the value of the car after 10 years?
To find \( V(10) \):
\[
V(10) = 24,250 \times (0.85)^{10}
\]
Calculating \( (0.85)^{10} \):
Using a calculator,
\[
(0.85)^{10} \approx 0.196874404
\]
Now, substituting this value back into the equation:
\[
V(10) = 24,250 \times 0.196874404 \approx 4,772.50
\]
Thus, the value of the car after 10 years is approximately
\[
\boxed{4,772.50}
\]
### 3. After how many years will the car be worth $20,000?
We want to find \( t \) such that \( V(t) = 20,000 \).
Setting up the equation:
\[
20,000 = 24,250 \times (0.85)^t
\]
First, divide both sides by 24,250:
\[
\frac{20,000}{24,250} = (0.85)^t
\]
Calculating \( \frac{20,000}{24,250} \):
\[
\frac{20,000}{24,250} \approx 0.8251
\]
Now, we have:
\[
0.8251 \approx (0.85)^t
\]
To solve for \( t \), we take the logarithm of both sides:
\[
\log(0.8251) = \log((0.85)^t)
\]
Using the power rule of logarithms:
\[
\log(0.8251) = t \cdot \log(0.85)
\]
Now solve for \( t \):
\[
t = \frac{\log(0.8251)}{\log(0.85)}
\]
Calculating the logarithms using a calculator:
\[
\log(0.8251) \approx -0.0822 \quad \text{and} \quad \log(0.85) \approx -0.0706
\]
Thus,
\[
t = \frac{-0.0822}{-0.0706} \approx 1.164
\]
Finally, rounding to the nearest tenth:
\[
t \approx 1.2
\]
So, the car will be worth $20,000 after about
\[
\boxed{1.2} \text{ years}.
\]