A car is purchased for $24,000 . Each year it loses 30% of its value. After how many years will the car be worth $7300 or less? (Use the calculator provided if necessary.) Write the smallest possible whole number answer.
Set up the depreciation equation D(t) where t is the number of years in the life of the car:
D(t) = 24,000 * (1 - 0.3)^t
D(t) = 24000 * (0.7)^t
The problem asks for D(t)<=7300
24000 * (0.7)^t = 7300
Divide each side by 24000
(0.7)^t = 7300/24000
(0.7)^t= 0.30416666666
Take the natural log of both sides:
LN(0.7^t) = -1.190179482215518
Using the natural log identities, we have:
t * LN(0.7) = -1.190179482215518
t * -0.35667494393873245= -1.190179482215518
Divide each side by -0.35667494393873245
t = 3.33687437943
Rounding this up, we have t = 4
Set up the depreciation equation D(t) where t is the number of years in the life of the car:
D(t) = 24,000 * (1 - 0.3)^t
D(t) = 24000 * (0.7)^t
The problem asks for D(t)<=7300
24000 * (0.7)^t = 7300
Divide each side by 24000
(0.7)^t = 7300/24000
(0.7)^t= 0.30416666666
Take the natural log of both sides:
LN(0.7^t) = -1.190179482215518
Using the natural log identities, we have:
t * LN(0.7) = -1.190179482215518
t * -0.35667494393873245= -1.190179482215518
Divide each side by -0.35667494393873245
t = 3.33687437943
Rounding this up, we have t = 4
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