Hey there!
A car is purchased for $23,000 . Each year it loses 30% of its value. After how many years will the car be worth $6700 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) = 23,000/(1.3)^t
The problem asks for D(t)<=6800
23,000/(1.3)^t = 6800
Cross multiply:
7300(1.3)^t = 24,000
Divide each side by 6800
1.3^t = 23000/6800
1.3^t = 3.38
Take the natural log of both sides:
LN(1.3^t) = LN(3.38)
Using the natural log identities, we have:
t * LN(1.3) = 1.22
t * 0.2624 = 1.22
Divide each side by 0.2624
t = 4.6494
Rounding this up, we have t = 5