Question

The value of a car is $27,300. It loses 12% of its value every year.

a. Write a function that represents the value y (in dollars) of the car after years.
y =
b. Estimate the value of the car after 5 years. Round to the nearest dollar.
About $

Answer 1

a) y=27300(0.880^x

b) $14,407

Explanation:

a. The function that represents the value y (in dollars) of the car after x years can be expressed using the following formula:

y = A(1-r)^x

where x is years in terms of the initial amount A, the annual depreciation rate r, and the number of years x:

y = 27300(1-12%)^x

y=27300(0.880^x is a required function.

b. To estimate the value of the car after 5 years, we can substitute x = 5 into the formula and evaluate it:

y = 27300(0.88)^5

y ≈ $14,407

Rounding to the nearest dollar, the estimated value of the car after 5 years is about $14,407.

Answer 2

`\textsf{a)} \quad y=27300 \cdot 0.88^x`

b) An estimate of the value of the car after 5 years rounded to the nearest dollar is $14,407.

Explanation:

a) If the car loses 12% of its value each year, we can use an exponential function to model its value in dollars after x years.

`\boxed{\begin{minipage}{9 cm}\underline{General form of an Exponential Function}\\\\$y=ab^x$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the initial value ($y$-intercept). \\ \phantom{ww}$\bullet$ $b$ is the base (growth/decay factor) in decimal form.\\\end{minipage}}`

The initial value of the car is $27,300. Therefore, a = 27300.

If the car loses 12% of its value every year, it will be 88% of the previous year's value, since 100% - 12% = 88%. Therefore, b = 0.88.

Substitute these values into the formula to create a function that represents the value y (in dollars) of the car after x years.

`\boxed{y=27300 \cdot 0.88^x}`

b) To estimate the value of the car after 5 years, substitute x = 5 into the function from part a.

`\implies y=27300 \cdot 0.88^5`

`\implies y=27300 \cdot 0.52773191...`

`\implies y=14407.081328...`

Therefore, an estimate of the value of the car after 5 years rounded to the nearest dollar is $14,407.