Question

Andrew researched the depreciation of cars and found that the value of the car he just bought halved every 5 years. Andrew purchased the car for $20,000 and he wants to find how many years it will take for the value of the car to reduce to $15,000. He wrote the equation 20000(5)^t/2=15000, and plans to solve for t. What change(s) should Andrew make to the equation to find the value of t in the above scenario? Use the half-life decay formula, m(t) = m o (1/2)^t/h, to answer the question.

A. t/2 should be replaced with 2t
B. 15000 and 20000 should be interchanged
C. 5 should be replaced with 1/2, and t/2 should be replaced with t/5
D. 5 should be replaced with 1/2, and t/2 should be replaced with 5t

Answer 1

C. 5 should be replaced with 1/2, and t/2 should be replaced with t/5

Explanation:

The problem statement tells you the half-life of the car is 5 years. Using the value h = 5 in the given equation, you have ...

m(t) = m₀(1/2)^(t/5)

The problem statement tells us the initial value of the car is $20,000, so we have ...

m₀ = 20000

m(t) = 20000(1/2)^(t/5)

Since Andrew wants the value of t when m(t) = 15000, his equation would read ...

15000 =20000(1/2)^(t/5) . . . equation for finding when the value is 15000

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The equation Andrew proposes to solve is ...

20000(5)^(t/2) = 15000

In order to make his equation look like the the one above, the changes should be ...

5 should be replaced with 1/2, and t/2 should be replaced with t/5.

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Additional comment

The equation can be solved as follows:

15000/20000 = (1/2)^(t/5)

log(3/4) = (t/5)log(1/2) . . . . take the log of both sides, simplify the fraction

t = 5log(3/4)/log(1/2) . . . . divide by the coefficient of t

t ≈ 2.08

Andrew's car will have a value of about $15,000 after 2 years and 1 month.