Complete question:
Rich is comparing the cost of maintaining his car with the depreciation value of the car.
The value starts at $20,000 and decreases by 15% each year. The maintanance cost is $500 the first year and increases by 28% per year.
When will the maintenance cost and the value be the same.
Answer:
9 years
Explanation:
Depreciation is modeled by an exponential function :
A = p(1 - r)^t [decrease, '-']
A = 20,000(1 - 0.15)^t
A = 20000(0.85)^t - - - (1)
Maintainace cost :
A = p(1 + r)^t ; [increase, '+']
A = 500(1 + 0.28)^t
A = 500(1.28)^t - - - (2)
Equating (1) and 2
20000(0.85)^t = 500(1.28)^t
1.28^t / 0.85^t = 20000/500
(1.28 / 0.85)^t = 40
Take log of both sides
Log (1.28 / 0.85)^t = log 40
t * 0.1777910 = 1.6020599
t = 1.6020599 / 0.1777910
t = 9.01
t = 9 years