Asked by Oakley
Timmy's mom bought a car for $9000 five years ago. She wants to sell it to Timmy based on a 15% annual rate of depreciation.What is the exponential function for this situation?
Answers
Answered by
GPT-4o mini
To express the depreciation of the car as an exponential function, we can use the formula for exponential decay:
\[
V(t) = V_0 \cdot (1 - r)^t
\]
Where:
- \(V(t)\) is the value of the car after \(t\) years,
- \(V_0\) is the initial value of the car,
- \(r\) is the depreciation rate,
- \(t\) is the time in years.
In Timmy's mom's case:
- The initial value \(V_0\) is $9000,
- The depreciation rate \(r\) is 15% or 0.15.
Now we can substitute these values into the formula:
\[
V(t) = 9000 \cdot (1 - 0.15)^t
\]
Simplifying this gives:
\[
V(t) = 9000 \cdot (0.85)^t
\]
So, the exponential function that describes the value of the car over time is:
\[
V(t) = 9000 \cdot (0.85)^t
\]
This function can be used to calculate the value of the car after any number of years \(t\).
\[
V(t) = V_0 \cdot (1 - r)^t
\]
Where:
- \(V(t)\) is the value of the car after \(t\) years,
- \(V_0\) is the initial value of the car,
- \(r\) is the depreciation rate,
- \(t\) is the time in years.
In Timmy's mom's case:
- The initial value \(V_0\) is $9000,
- The depreciation rate \(r\) is 15% or 0.15.
Now we can substitute these values into the formula:
\[
V(t) = 9000 \cdot (1 - 0.15)^t
\]
Simplifying this gives:
\[
V(t) = 9000 \cdot (0.85)^t
\]
So, the exponential function that describes the value of the car over time is:
\[
V(t) = 9000 \cdot (0.85)^t
\]
This function can be used to calculate the value of the car after any number of years \(t\).