Question

1. your car cost $42,500 when you purchased it in 2015. the value of the car decreases by 15% annually.

a. write an exponential decay function to represent this situation.
b. how much will your car be worth in 2022? round your answer to the nearest dollar.

1. a piece of land was purchased for $65,000. the value of the land has slowly been decreasing by 1% annually.

a. write an exponential decay function to represent this situation.
b. how much will the land be worth in 20 years? round your answer to the nearest dollar.
© pecktabo math 2015

Explanation:

(Question 15):

Step1: Define decay function form

The general exponential decay function is $y = a(1-r)^x$, where $a$ = initial value, $r$ = decay rate, $x$ = time in years.

Step2: Plug in car's values

Initial cost $a = 42500$, decay rate $r = 0.15$.
Function: $y = 42500(1-0.15)^x = 42500(0.85)^x$

Step3: Calculate time elapsed

2022 - 2015 = 7 years, so $x=7$.

Step4: Compute car's 2022 value

$y = 42500(0.85)^7$
First calculate $(0.85)^7 \approx 0.320577$
Then $y \approx 42500 \times 0.320577 \approx 13624.52$

(Question 16):

Step1: Define decay function form

Use $y = a(1-r)^x$, where $a$ = initial value, $r$ = decay rate, $x$ = time in years.

Step2: Plug in land's values

Initial cost $a = 65000$, decay rate $r = 0.01$.
Function: $y = 65000(1-0.01)^x = 65000(0.99)^x$

Step3: Compute land's value at x=20

$y = 65000(0.99)^{20}$
First calculate $(0.99)^{20} \approx 0.817908$
Then $y \approx 65000 \times 0.817908 \approx 53164.02$

Answer:

Question 15

A. $y = 42500(0.85)^x$
B. $\$13625$

Question 16

A. $y = 65000(0.99)^x$
B. $\$53164$