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)=26,000(0.88)^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: $\square$ value after 12 years: $\square$

Explanation:

Step1: Find Initial Value

The initial value of an exponential function \( v(t) = a(b)^t \) is when \( t = 0 \). Substitute \( t = 0 \) into \( v(t)=26000(0.88)^t \). Since any non - zero number to the power of 0 is 1, \( v(0)=26000\times(0.88)^0 = 26000\times1=26000 \).

Step2: Find Value after 12 Years

Substitute \( t = 12 \) into the function \( v(t)=26000(0.88)^t \). So we need to calculate \( v(12)=26000\times(0.88)^{12} \). First, calculate \( (0.88)^{12}\approx0.215671 \) (using a calculator). Then multiply by 26000: \( 26000\times0.215671\approx5607.45 \).

Answer:

Initial value: \(\$26000\)
Value after 12 years: \(\$5607\) (rounded to the nearest dollar)