Final answer:
The value of the backhoe after 3 years is $41,164.05, after 6 years is $29,986.76, and after 10 years is $19,657.06, based on an annual depreciation rate of 10% modeled by an exponential decay function.
Step-by-step explanation:
Calculating Depreciation Using Exponential Decay
To calculate the value of the backhoe over time, we use the given exponential function v(t) = 56,395 (0.9)t, where t represents the number of years since the backhoe was purchased. The function models depreciation at a rate of 10% per year, as the value is 90% of the previous year's each year.
Let's calculate the value after 3 years first:
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- v(3) = 56,395 × (0.9)3
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- v(3) = 56,395 × 0.729
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- v(3) = 41,164.05
The value of the backhoe after 3 years is $41,164.05.
Now, for the value after 6 years:
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- v(6) = 56,395 × (0.9)6
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- v(6) = 56,395 × 0.531441
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- v(6) = 29,986.76
The value after 6 years is $29,986.76.
Finally, let's find the value after 10 years:
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- v(10) = 56,395 × (0.9)10
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- v(10) = 56,395 × 0.3486784401
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- v(10) = 19,657.06
After 10 years, the backhoe will have a value of $19,657.06.
This calculation demonstrates how exponential decay can be used to model the loss of value of an asset such as a backhoe over time due to depreciation.