Ethan purchases a car for $25,000, and the value depreciates by half every 3 years. The time t, in years, that it takes for the value of a car to reach an amount P

Question

𝑃
, in dollars, is calculated by the logarithmic equation t=3log2(25,000P)
𝑡
=
3
log
2
⁡
(
25
,
000
𝑃
)
. Which of the following is the best estimate for the value of the car after 5 years?(1 point)
Responses

$10,000
$10,000

$8,000
$8,000

$6,000
$6,000

$5,000

Answer 1

To estimate the value of the car after 5 years, we can use the information that the car's value depreciates by half every 3 years. This means that:

• After 3 years, the value is $\frac{25,000}{2}=12,500$.
• After another 2 years (which is 5 years total), we need to find the value of the car.

From year 3 to year 5 (which is a period of 2 years), the value of the car will not halve again, but it will continue to depreciate.

To find the value after 5 years, we can estimate that the car's value decreases in a linear way between the intervals. We know that after every 3 years the value halves, and within the remaining 2 years, it will depreciate further but not as significantly.

To simplify, we can say that about two-thirds of the time interval from 3 years to 6 years means a depreciation of approximately one-third of the total depreciation that would happen in 3 years. After 3 years, the value was $12,500. Therefore in the next 2 years, we can estimate its depreciation:

1. Value at 3 years: $12,500
2. Estimated depression in 2 years (from 3 years to 6 years):
   • One-third of remaining depreciation $=\frac{1}{3}\times \frac{12,500}{2}=\frac{1}{3}\times 6,250=2,083.33$
3. Estimated value after another 2 years:
   • $12,500-2,083.33\approx 10,416.67$

Considering the actual answers provided, the closest estimate to the value of the car after 5 years would be around $10,000.

Therefore, the best estimate for the value of the car after 5 years is $10,000.