Question
Leo's Mom purchased a car for $32,000. The value of her car decreases by 14% annually. You can use a function to describe the value of the car x years since Leo's mom bought it.
Match the below with the correct answer.
(3 points)
Questions:
Is this Linear or Exponential?
What formula would you need to solve this type of equation? f(x)=32000(1-0.14)^2 or f(x)=32000(1+0.14)^2?
After 2 years what would the formula for depreciation be on Leo's mom's Car? y=mx+b, f(x)=a(1+r)^t, or f(x)a(1-r)^t?
What is the value of her car after 2 years? $23,667.00 or $41,587.20?
Match the below with the correct answer.
(3 points)
Questions:
Is this Linear or Exponential?
What formula would you need to solve this type of equation? f(x)=32000(1-0.14)^2 or f(x)=32000(1+0.14)^2?
After 2 years what would the formula for depreciation be on Leo's mom's Car? y=mx+b, f(x)=a(1+r)^t, or f(x)a(1-r)^t?
What is the value of her car after 2 years? $23,667.00 or $41,587.20?
Answers
GPT-4o mini
Let's break down the questions one by one based on the information provided.
1. **Is this Linear or Exponential?**
The value of the car decreases by a percentage each year, which indicates an exponential decay. Therefore, the answer is **Exponential**.
2. **What formula would you need to solve this type of equation?**
To calculate the value of the car after a certain number of years with a decrease of 14%, we use the formula for exponential decay:
\( f(x) = a(1 - r)^t \)
where \( a \) is the initial amount, \( r \) is the rate of decrease, and \( t \) is the number of years. With the given options, the correct match would be **f(x) = 32000(1 - 0.14)^t** (if listing options, ensure the correct exponent reflects the number of years).
3. **After 2 years what would the formula for depreciation be on Leo's mom's Car?**
The correct formula for calculating the value of the car based on depreciation is indeed the exponential decay formula. Therefore, the correct answer is **f(x) = a(1 - r)^t** (specifically, in this context, we'd use \( f(x) = 32000(1 - 0.14)^t \)).
4. **What is the value of her car after 2 years?**
To find the value after 2 years, you can substitute \( t = 2 \) into the formula.
\[
f(2) = 32000(1 - 0.14)^2 = 32000(0.86)^2
\]
\[
f(2) = 32000 \times 0.7396 \approx 23667.20
\]
Thus, the value of the car after 2 years is approximately **$23,667.00**.
### Summary:
1. Exponential
2. f(x) = 32000(1 - 0.14)^t
3. f(x) = a(1 - r)^t
4. $23,667.00
1. **Is this Linear or Exponential?**
The value of the car decreases by a percentage each year, which indicates an exponential decay. Therefore, the answer is **Exponential**.
2. **What formula would you need to solve this type of equation?**
To calculate the value of the car after a certain number of years with a decrease of 14%, we use the formula for exponential decay:
\( f(x) = a(1 - r)^t \)
where \( a \) is the initial amount, \( r \) is the rate of decrease, and \( t \) is the number of years. With the given options, the correct match would be **f(x) = 32000(1 - 0.14)^t** (if listing options, ensure the correct exponent reflects the number of years).
3. **After 2 years what would the formula for depreciation be on Leo's mom's Car?**
The correct formula for calculating the value of the car based on depreciation is indeed the exponential decay formula. Therefore, the correct answer is **f(x) = a(1 - r)^t** (specifically, in this context, we'd use \( f(x) = 32000(1 - 0.14)^t \)).
4. **What is the value of her car after 2 years?**
To find the value after 2 years, you can substitute \( t = 2 \) into the formula.
\[
f(2) = 32000(1 - 0.14)^2 = 32000(0.86)^2
\]
\[
f(2) = 32000 \times 0.7396 \approx 23667.20
\]
Thus, the value of the car after 2 years is approximately **$23,667.00**.
### Summary:
1. Exponential
2. f(x) = 32000(1 - 0.14)^t
3. f(x) = a(1 - r)^t
4. $23,667.00