Answer :
Certainly! To determine the value of the car after 3 years, given that it depreciates by 5% each year, we can follow these steps:
### Step-by-Step Solution
1. Understand Depreciation: Depreciation means the car loses value each year. A 5% depreciation rate implies the car retains 95% (100% - 5%) of its value each year.
2. Write the Exponential Function:
- In general, the formula for depreciation can be written as:
[tex]\[ V(t) = P \times (1 - r)^t \][/tex]
where:
- [tex]\(V(t)\)[/tex] is the value of the car after [tex]\(t\)[/tex] years,
- [tex]\(P\)[/tex] is the initial principal value (purchase price),
- [tex]\(r\)[/tex] is the annual depreciation rate,
- [tex]\(t\)[/tex] is the number of years.
- For this problem:
[tex]\[ V(t) = 15000 \times (1 - 0.05)^t \][/tex]
Which simplifies to:
[tex]\[ V(t) = 15000 \times (0.95)^t \][/tex]
3. Calculate the Car's Value After 3 Years:
- Plug in [tex]\(t = 3\)[/tex] into the function:
[tex]\[ V(3) = 15000 \times (0.95)^3 \][/tex]
4. Perform the Calculation:
- Compute [tex]\((0.95)^3\)[/tex]:
[tex]\[ (0.95)^3 = 0.857375 \][/tex]
- Multiply by the principal value:
[tex]\[ V(3) = 15000 \times 0.857375 = 12860.625 \][/tex]
So, the value of the car 3 years later is $12,860.63 (rounded to two decimal places).
### Step-by-Step Solution
1. Understand Depreciation: Depreciation means the car loses value each year. A 5% depreciation rate implies the car retains 95% (100% - 5%) of its value each year.
2. Write the Exponential Function:
- In general, the formula for depreciation can be written as:
[tex]\[ V(t) = P \times (1 - r)^t \][/tex]
where:
- [tex]\(V(t)\)[/tex] is the value of the car after [tex]\(t\)[/tex] years,
- [tex]\(P\)[/tex] is the initial principal value (purchase price),
- [tex]\(r\)[/tex] is the annual depreciation rate,
- [tex]\(t\)[/tex] is the number of years.
- For this problem:
[tex]\[ V(t) = 15000 \times (1 - 0.05)^t \][/tex]
Which simplifies to:
[tex]\[ V(t) = 15000 \times (0.95)^t \][/tex]
3. Calculate the Car's Value After 3 Years:
- Plug in [tex]\(t = 3\)[/tex] into the function:
[tex]\[ V(3) = 15000 \times (0.95)^3 \][/tex]
4. Perform the Calculation:
- Compute [tex]\((0.95)^3\)[/tex]:
[tex]\[ (0.95)^3 = 0.857375 \][/tex]
- Multiply by the principal value:
[tex]\[ V(3) = 15000 \times 0.857375 = 12860.625 \][/tex]
So, the value of the car 3 years later is $12,860.63 (rounded to two decimal places).
