Question

nora bought a motorcycle for $12,000. one year later, she looked online and found that the value of her motorcycle had decreased to about $10,560. based on this online research, she expects the motorcycle to continue decreasing in value each year.
write an exponential equation in the form $y = a(b)^x$ that can model the value of nora’s motorcycle, $y$, $x$ years after purchase.
use whole numbers, decimals, or simplified fractions for the values of $a$ and $b$.
$y = square$
to the nearest ten dollars, how much can nora expect her motorcycle to be worth 5 years after purchase?
$ square$

Explanation:

Step1: Identify initial value $a$

The initial purchase price is $a = 12000$.

Step2: Calculate decay factor $b$

Divide the value after 1 year by initial value:
$b = \frac{10560}{12000} = 0.88$

Step3: Write exponential equation

Substitute $a$ and $b$ into $y=a(b)^x$:
$y = 12000(0.88)^x$

Step4: Calculate value at $x=5$

Substitute $x=5$ into the equation:
$y = 12000(0.88)^5$
First compute $(0.88)^5 = 0.88\times0.88\times0.88\times0.88\times0.88 \approx 0.5277319168$
Then $y \approx 12000\times0.5277319168 = 6332.7829996$

Step5: Round to nearest ten dollars

Round $6332.78$ to the nearest ten: $6330$

Answer:

Exponential equation: $y = 12000(0.88)^x$
Value after 5 years: $\$6330$