Question

The value v of a certain automobile that is t years old can be modeled by upper v left parenthesis t right parenthesis equals 14 comma 683 left parenthesis 0.78 right parenthesis superscript tv(t)=14,683(0.78)t. according to the​ model, when will the car be worth ​(a) ​$90009000​? ​(b) ​$30003000​? ​(c) ​$20002000​?

Answer 1

The given equation is as:

$v(t)=14683(0.78)^t$

We will use this equation in all the three cases to find the $t$ in years.

Case (a)

Since $v(t)=9000$

Our equation will become

$9000=14683(0.78)^t$

Dividing 14683 on both sides and taking log to the base 10 on both sides, we will get:

$log(\frac{9000}{14683})=t\times log(0.78)$

Thus, $t=\frac{log(\frac{9000}{14683})}{log(0.78)}\approx1.97$ year

Case (b)

Since $v(t)=3000$

Our equation will become

$3000=14683(0.78)^t$

Dividing 14683 on both sides and taking log to the base 10 on both sides, we will get:

$log(\frac{3000}{14683})=t\times log(0.78)$

Thus, $t=\frac{log(\frac{3000}{14683})}{log(0.78)}\approx6.4$ years

Case (C)

Since $v(t)=2000$

Our equation will become

$2000=14683(0.78)^t$

Dividing 14683 on both sides and taking log to the base 10 on both sides, we will get:

$log(\frac{2000}{14683})=t\times log(0.78)$

Thus, $t=\frac{log(\frac{2000}{14683})}{log(0.78)}\approx8$ years