Question
Based on her past experiences, a homeowner estimates that appliances lose 25% of their resale value each year. If her estimate is accurate, how much will an appliance currently valued at $1,500 be worth in 8 years?
If necessary, round your answer to the nearest cent.
If necessary, round your answer to the nearest cent.
Answers
GPT-4o mini
To calculate the future value of the appliance after 8 years, given that it loses 25% of its value each year, we can use the formula for exponential decay:
\[
V = P \times (1 - r)^t
\]
Where:
- \( V \) is the future value of the appliance.
- \( P \) is the current value of the appliance ($1,500).
- \( r \) is the rate of depreciation (25% or 0.25).
- \( t \) is the number of years (8 years).
Now, let's plug in the values:
\[
V = 1500 \times (1 - 0.25)^8
\]
This simplifies to:
\[
V = 1500 \times (0.75)^8
\]
Now, we calculate \( (0.75)^8 \):
\[
(0.75)^8 \approx 0.100112915
\]
Next, we multiply this result by $1,500:
\[
V \approx 1500 \times 0.100112915 \approx 150.1683723
\]
Finally, rounding to the nearest cent, the worth of the appliance after 8 years will be approximately:
\[
\text{Value} \approx \text{\$150.17}
\]
Therefore, the appliance will be worth **$150.17** in 8 years.
\[
V = P \times (1 - r)^t
\]
Where:
- \( V \) is the future value of the appliance.
- \( P \) is the current value of the appliance ($1,500).
- \( r \) is the rate of depreciation (25% or 0.25).
- \( t \) is the number of years (8 years).
Now, let's plug in the values:
\[
V = 1500 \times (1 - 0.25)^8
\]
This simplifies to:
\[
V = 1500 \times (0.75)^8
\]
Now, we calculate \( (0.75)^8 \):
\[
(0.75)^8 \approx 0.100112915
\]
Next, we multiply this result by $1,500:
\[
V \approx 1500 \times 0.100112915 \approx 150.1683723
\]
Finally, rounding to the nearest cent, the worth of the appliance after 8 years will be approximately:
\[
\text{Value} \approx \text{\$150.17}
\]
Therefore, the appliance will be worth **$150.17** in 8 years.