數學題5.........

The value of a second-hand car reduces exponentially with age, so that its value $y after t years can be modeled by the formula y=Aa^t

If the car was $50000 when new and was worth $38000 after two years, find the values of A and a , correct to three decimal places.
a. Use this model to predict the value of the car (i) when the car is 5 years old (ii) in the long run.
b. Find the half life of the car.
a
when t=0 y=50000
when t=2 y=38000
A=50000
Aa^2=38000
50000a^2=38000
a=0.872
so
y=50000(0.872)^t
(i) when the car is 5 years old
the value is
50000(0.872)^5
=28909.191
(ii) in the long run
since (0.872)^t tends to 0 as t tends to infinity
so in the long run, the value is 0
let
50000(0.872)^t=1/2(50000)
(0.872)^t=0.5
tln0.872=ln0.5
t=5.061
the half life of the car is 5.061 years

very good!!!