College algebra

Question

 A car’s value varies inversely with its age. Let’s say a new car costs $20,000 and eight years later it is worth $5,000. Come up with the function representing the value decrease over time, and then calculate how much will the car be worth after 12 years.

Answer 1

P=20e^rt where t = years, r= annual percentage change in value, 20 =original value in thousands of dollars. P= new value after t years

5 = 20e^8r

5/20 = e^8r

1/4 = e^8r

take natural logs of both sides

ln(1/4) = 8r

r = ln.25/8 = -1.3863/8 = -0.1733

12r = -0.1733(12) = -2.0794

A = 20e^-2.0794t =20e^-2.0794(12) = 20(.125)= 2.5 thousand dollars

Value after 12 years is $2,500

or

it lost 3/4 of its value in 8 years. 5 is 1/4 of 20 in 8 years

in 4 years it's lost 1/2 it's value

K=$1,000

20K in 0 years

10K in 4 years

5K in 8 years

2.5K in 12 years

every 4 years it loses 1/2 its value

P = A(1/2)^T where T = number of 4 year periods. 12 years = 3 four year periods. T=3

P =20(1/2)^3 = 20(1/8 = 20/8 = 2.5 thousand dollars

or

P =20(1/2)^t/4

P =20(1/2)^12/4 = 20(1/2)^3 = 20/8 = 2.5 thousand dollars