Question

A bicycle costs $240 and it loses 3/5 of its value each year.

A. Write expressions for the value of the bicycle,in dollars, after 1,2, and 3 years

B. When will the bike be worth less than $1?

Answer 1

A. $96.00, $38.40, $15.36

B. after 6 years

Explanation:

An exponential function has the general form ...

y = (initial value) × (growth factor)^t

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A.

The growth factor is 1 more than the growth rate. Here, the "growth" is actually a decay at the rate of 60% per year, so the growth rate is -0.60. The initial value is $240. The value equation can be written ...

y = 240(1 -0.60)^t

We want the value after t=1, t=2, and t=3 years.

y = $240(0.4)^1 = $96 . . . value after 1 year

y = $240(0.4)^2 = $38.40 . . . value after 2 years

y = $240(0.4)^3 = $15.36 . . . value after 3 years

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B.

We want to find t when y < 1.

240(0.4)^t < 1

0.4^t < 1/240 . . . . . . . . . . . . divide by 240

t × log(0.4) < log(1/240) . . . take logs

t > log(1/240)/log(0.4) . . . . . . . inequality reverses because log(0.4) < 0

t > 5.98

The bike will be worth less than $1 after about 6 years.

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Additional comment

For part A, we have shown the values after 1, 2, and 3 years. The expressions are shown on the same line preceding the value.