High School

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2. If a boat depreciates in value according to the formula [tex]y = 20000(0.92)^t[/tex], find the rate of decay.

Show your work and write your answer as a percentage.

Answer :

To find the rate of decay for the boat's value according to the formula [tex]\( y = 20000(0.92)^t \)[/tex], we start by recognizing that the equation is an example of exponential decay. In this context:

- [tex]\( A = 20000 \)[/tex] is the initial value of the boat.
- [tex]\( 0.92 \)[/tex] is the decay factor, which represents the fraction of the boat's value remaining after each year.

To determine the rate of decay, follow these steps:

1. Identify the decay factor in the equation, which is [tex]\( 0.92 \)[/tex]. This represents the value of the boat after one year as a percentage of its previous value.

2. Calculate the rate of decay using the formula:
[tex]\[
\text{Rate of decay} = 1 - \text{Decay factor}
\][/tex]

3. Substitute the decay factor value into the formula:
[tex]\[
\text{Rate of decay} = 1 - 0.92 = 0.08
\][/tex]

4. Convert the rate of decay into a percentage:
[tex]\[
\text{Rate of decay percentage} = 0.08 \times 100 = 8\%
\][/tex]

So, the rate of decay of the boat's value is 8% per year. This means that each year, the value of the boat decreases by 8% compared to the previous year.

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