the dollar value v(t) of a certain car model that is t years old is given by the following exponential…

the dollar value v(t) of a certain car model that is t years old is given by the following exponential function. v(t)=24,500(0.84)^t. find the initial value of the car and the value after 12 years. round your answers to the nearest dollar as necessary. initial value: $ value after 12 years: $

the dollar value v(t) of a certain car model that is t years old is given by the following exponential function. v(t)=24,500(0.84)^t. find the initial value of the car and the value after 12 years. round your answers to the nearest dollar as necessary. initial value: $ value after 12 years: $

Answer

Explanation:

Step1: Find the initial value

The general form of an exponential - decay function is $v(t)=v_0r^t$, where $v_0$ is the initial value and $r$ is the decay factor. In the given function $v(t) = 24500(0.84)^t$, when $t = 0$, we have $v(0)=24500(0.84)^0$. Since any non - zero number to the power of 0 is 1, $v(0)=24500$. So the initial value of the car is $$24500$.

Step2: Find the value after 12 years

Substitute $t = 12$ into the function $v(t)=24500(0.84)^t$. Then $v(12)=24500\times(0.84)^{12}$. Calculate $(0.84)^{12}\approx0.1469$. Then $v(12)=24500\times0.1469 = 3609.05\approx3609$.

Answer:

Initial value: $$24500$ Value after 12 years: $$3609$