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Difficulty: MediumLinear and Exponential Growth

A consumer electronics reviewer is tracking the resale value of two different smartphone models. At the time of purchase (t=0t = 0 years), Smartphone A and Smartphone B each have a retail value of 1,0001,000 dollars. The value of Smartphone A decreases by 10%10\% of its value from the previous year each year. The value of Smartphone B decreases by a constant amount of 100100 dollars each year. What is the difference, in dollars, between the values of the two smartphones 33 years after they were purchased?

  1. A
    0
  2. B
    20
  3. C
    429
  4. 29Answer

Answer

The difference between the values of the two smartphones is 29 dollars.
The correct answer is the option showing a difference of 2929 dollars. To solve this, we model each smartphone's resale value over time. Smartphone A experiences exponential decay with a decay factor of 0.900.90. Its value after 33 years is 1,000(0.90)3=7291,000(0.90)^3 = 729 dollars. Smartphone B experiences linear decay with a constant rate of decrease of 100100 dollars per year. Its value after 33 years is 1,000−100(3)=7001,000 - 100(3) = 700 dollars. The difference between these two values is 729−700=29729 - 700 = 29 dollars.

Step-by-Step Solution

1
Find the value of Smartphone A at t=3t = 3 years.
The resale value of Smartphone A is 1,000(1−0.10)3=1,000(0.90)3=1,000(0.729)=7291,000(1 - 0.10)^3 = 1,000(0.90)^3 = 1,000(0.729) = 729 dollars.
Smartphone A decreases by a constant percentage of 10%10\% each year, which represents exponential decay.
2
Find the value of Smartphone B at t=3t = 3 years.
The resale value of Smartphone B is 1,000−100(3)=1,000−300=7001,000 - 100(3) = 1,000 - 300 = 700 dollars.
Smartphone B decreases by a constant absolute amount of 100100 dollars each year, which represents linear decay.
3
Calculate the difference between the two values at t=3t = 3 years.
729−700=29729 - 700 = 29 dollars.
Subtract the smaller value from the larger value to find the positive difference.

Key Concept

Linear and Exponential Growth and Decay
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Topics

Linear and Exponential Growth

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