Exponential Decay Quick Check

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Question
A local bakery opened with new equipment worth $80,000. Ten years later, the owners’ accountants told them that their equipment had depreciated at a rate of 6 percent per year. Which equation would you use to create a sequence modeling this situation?(1 point)
Responses

an=80,000(0.06)n−1
a subscript n baseline equals 80,000 left parenthesis 0.06 right parenthesis superscript n minus 1 baseline

an=80,000(0.94)n
a subscript n baseline equals 80,000 left parenthesis 0.94 right parenthesis superscript n baseline

an=80,000(0.06)n
a subscript n baseline equals 80,000 left parenthesis 0.06 right parenthesis superscript n baseline

an=80,000(0.94)n−1
a subscript n baseline equals 80,000 left parenthesis 0.94 right parenthesis superscript n minus 1 baseline
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Answer 1

To model the exponential decay of the bakery equipment's value over time, we need to consider that the equipment depreciates at a rate of 6% per year.

The value of the equipment after \( n \) years can be modeled with the formula:

\[ a_n = a_0 (1 - r)^n \]

where:

• \( a_0 \) is the initial value (in this case, $80,000),
• \( r \) is the rate of depreciation (0.06 for 6%),
• \( n \) is the number of years.

In this scenario, the decay factor becomes \( 1 - r = 1 - 0.06 = 0.94 \).

Thus, the equation modeling the situation will be:

\[ a_n = 80,000 (0.94)^n \]

So, the correct response to select is:

an=80,000(0.94)n

This indicates the value of the equipment after \( n \) years at a rate of depreciation of 6% per year.