initial value? value after 12 years

Question

The dollar value v(t) of a certain car model that is t years old is given by the following exponential function. v(t)=32,000(0.80)^t .Find the initial value of the car and the value after 12 years. Round your answers to the nearest dollar as necessary.

Answer 1

Hello Lily!

With exponential functions of this form, the initial value will often be the leading coefficient. i.e. in your problem where v(t)=32,000(0.80)^t, 32,000 would be the initial value as I have underlined. This is because the initial value occurs at t = 0, and we know that any number raised to the zero power = 1. So (0.80)^0 = 1, and 32,000*1 = 32,000 = v(0).

As far as the value after 12 years, we repeat the process, but with t = 12. i.e. --> (0.80)^12 = .0687, and 32,000*.0687 = 2,199 = v(12). Therefore the value after 12 years is $2,199.

Hopefully this helps!

Aaron