Q7. - Algebra 1

Question

Record your answers on the answer sheet provided by your teacher or on a sheet of paper.

Misty purchased a car several years ago $\$21459$. The value of the car depreciated at a rate of $15\%$ annually. What was the value of the car after 5 years? Round your answer to the nearest whole dollar.

 

Step-by-Step Solution

Answer

 The value of the car after 5 years after rounding to the nearest whole dollar is $\mathit{\$}\mathbf{9521}$. 

Step 1. Write the equation for the exponential decay.

The equation for the exponential decay is given by:

 $y=a{\left(1-r\right)}^t$, where $y$ is the final amount, $a$ is the initial amount, $r$ is the rate of decay expressed as a decimal, and $t$ is the time.

Step 2. Find the value of the car after 5 years.

It is given that the initial value of the car is $\$21459$ . Therefore, the value of  $a$ is 21459.

The rate of depreciation is $15\%$. Therefore, the value  expressed as a decimal is given by:

 $$\begin{gathered} 15\%=\frac{15}{100} \\ =0.15 \end{gathered}$$

The given time is 5 years. Therefore, the value of $t$ is 5.

Now, substitute the values of $a, r$ and $t$ in the equation $y=a{\left(1-r\right)}^t$.

Therefore, it is obtained that:

 $y=a{\left(1-r\right)}^t$

   $$\begin{gathered} =21459{\left(1-0.15\right)}^5 \\ =21459{\left(0.85\right)}^5 \\ =21459\left(0.4437053125\right) \\ =9521.4723009375 \end{gathered}$$

Therefore, the value of the car after 5 years is $\$9521.4723009375$.

Therefore, the value of the car after 5 years after rounding to the nearest whole dollar is $\$9521$.