The value of a new car is $25,000. Each year, the value of the car is estimated to decrease by of its value at the beginning of that year.
Which equation models the estimated value , in dollars, of the car after years, where ?
A repeated percent decrease from the current value is exponential decay: each year multiplies the amount left by the same factor. Find the percent that remains after one year, then use its decimal form as the yearly factor. Put the starting value in front and raise the factor to the number of years. Using the percent lost as the factor would keep the wrong portion.
Hints
- Hint 1
A percent decrease tells you how much is taken away, not how much is left. Start with the whole car value as . After the yearly loss, what percent of that year's starting value remains?
- Hint 2
In an exponential model, the same factor is applied repeatedly. The car has a starting value, and counts how many yearly changes occur. Which number belongs in front, and which factor should be raised to ?
Step-by-step
Find the yearly factor
Step 1Find the percent left each year
No Desmos needed. The yearly factor comes straight from the percent decrease. The whole value at the start of a year is . Subtract the lost, then write the percent remaining as a decimal:
- Step 2
Apply the factor to the current value
The loss is based on the value at the beginning of that year, not on the original price every time. So if is the value after years, the next year's value is:
The yearly factor is what remains, not what was lost.
- Step 3
Write the model for years
The starting value is $25,000. After years, the factor has been applied times, so:
This models the car's estimated value after years. Choice B.