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

Question:

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

{eq}v(t) = 20,000(0.90)^t {/eq}

Find the initial value of the car and the value after 11 years.

Function:

A function is a special type of binary relation between two sets, in which one the terms of one set associate exactly with one term of another set. The given exponential function defines the value of a car after t years. We can find the value when we substitute the value of the year in the given function.

Answer and Explanation: 1

{eq}\begin{align*} v(t) &= 20000(0.90)^t\\[0.3 cm] v(0)&=20000(0.90)^0&&\text{[Substitute the value of x as 0 for the initial value of the car]}\\[0.3 cm] v(0)&=20000(1)&&\text{[Simplify]}\\[0.3 cm] v(0)&=$20000&&\text{[Simplify]}\\[0.3 cm] v(11)&=20000(0.90)^{11}&&\text{[Substitute the value of x as 11 for the value of the car after 11 years]}\\[0.3 cm] v(11)&\approx 20000(0.313810596)&&\text{[Simplify]}\\[0.3 cm] v(11)&\approx $6276.21 \end{align*} {/eq}

Therefore, the initial value of the car is {eq}$20,000 {/eq} and the value of the car after 11 years is {eq}$6276.21 {/eq}.


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