Question Find the camper van's value $y$ after $x$ years using the exponential equation $y = \$38,000 \cdot (0.85)^x$ .

Studdy Solution

STEP 1

1. The initial value of the camper van is $\$38,000$ .
2. The camper van depreciates at a constant rate of $15\%$ per year.
3. The exponential equation will be in the form $y = a(b)^x$ , where $y$ is the value of the van after $x$ years, $a$ is the initial value, and $b$ is the base of the exponential function representing the annual depreciation rate.

STEP 2

1. Determine the initial value $a$ of the van.
2. Calculate the annual depreciation rate $b$ .
3. Write the exponential equation using the values of $a$ and $b$ .

STEP 3

Identify the initial value $a$ of the van, which is the value at time $x=0$ .
$a = \$38,000$

STEP 4

Calculate the annual depreciation rate $b$ . Since the van depreciates by $15\%$ each year, it retains $85\%$ of its value each year. To represent this as a decimal, divide $85$ by $100$ .
$b = \frac{85}{100} = 0.85$

STEP 5

Write the exponential equation using the values of $a$ and $b$ .
$y = a(b)^x$ $y = 38000(0.85)^x$
The exponential equation that models the value of the van, $y$ , in $x$ years is:
$y = 38000(0.85)^x$

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