Chapter 6.4, Problem 32E

To determine

To find: the error in given solution.

Answer to Problem 32E

The error is in the model for exponential decay function for the given data.

Correct function is $v\left(t\right)=25,000{\left(0.86\right)}^t$

  $v\left(5\right)\approx \$11,760$

Explanation of Solution

Given:

The value of car decreases by 14% annually. The purchase value of car in 2010 is $25,000.

Then, value of car in2015 is:

  $\begin{array}{c}v\left(t\right)=25,000{\left(1.14\right)}^t\\ v\left(5\right)=25,000{\left(1.14\right)}^5\\ \approx 48,135\end{array}$

So, value of car in 2015 is $48,000.

Concept used:

Exponential decay function:

  $y=a{\left(1-r\right)}^t$ ….. (1)

Here, the y is the final value, a is the initial value, and r is the rate of decay (in decimal form).

Calculation:

The error here is in the model for exponential growth function for the given data.

Here, $a=25,000$ and $r=14\%=0.14$

So, the correct function is

  $\begin{array}{c}v\left(t\right)=a{\left(1-r\right)}^t\\ =25,000{\left(1-0.14\right)}^t\\ =25,000{\left(0.86\right)}^t\end{array}$

Then, value of car in 2015 is:

  $\begin{array}{c}v\left(5\right)=25,000{\left(1.14\right)}^5\\ \approx 11760.675\end{array}$

So, value of car in 2015 will be about $11,760.