A car losing value: an exponential model

Name: ______________________ Date: ____________ A car losing value: an exponential model (6 points)

Table

A used car loses value over time. Its value is

$24{,}000

one year after it was bought new and

$14{,}500

four years after it was bought. The value of the car, in dollars, is modeled by the function

C

given by

C(t) = ab^t

, where

t

is the number of years since the car was bought new.

Value of the car, in dollars, t years after it was bought new
$t$ (years)	1	4
Value (dollars)	24,000	14,500

(A(i)) Use the given data to write two equations that can be used to find the values of the constants $a$ and $b$ in the expression for $C(t)$ . [1 point]
(A(ii)) Find the values of $a$ and $b$ as decimal approximations. [1 point]
(B(i)) Find the average rate of change of $C$ over the interval $1 \le t \le 4$ . Express your answer using correct units. [1 point]
(B(ii)) Use the average rate of change from part B(i) to estimate the value of the car at $t = 3$ years. Show the work that leads to your answer. [1 point]
(C) Is the estimate in part B(ii) an overestimate or an underestimate of $C(3)$ ? Give a reason for your answer, using the concavity of the graph of $C$ on the interval $1 \le t \le 4$ . [2 points]

The question