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My question is as follows:

1.) An earth compactor costs $38,000 and has an economic life of 9 years. However, the purchaser needs it for only 1 project that will be completed in 3 years. At the end of the project, it can be sold for 1/2 its purchase price. What is the annual cost to the owner if the required rate of return is 12%? Economics for Engineers has been the bane of my existence.

Thank you so much for any help you can provide.

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Let $P\ $ ($=\$\ 38'000$) be the initial price of the machine and $p$ ($=0.12$) be the interest rate. At the beginning the purchaser has to take the amount $P$ out of his bank account, so after three years the total amount $(1+p)^3 P\ $ is missing there. On the other hand from the sale of the machine he then can pay back the amount ${P\over 2}$. It follows that the total cost per year is $${1\over3}\bigl((1+p)^3-{1\over2}\bigr) P\ ,$$ which amounts to $11'462$ dollars and $42$ cents.

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At the end of 3 years, it can be sold for $19000$. So $$P \left(\frac{(1.12)^{3}-1}{0.12} \right) = 19000$$ So $P \approx 5630$.

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If the loss on the machine is 19000, the cost per year (with no interest) would be 19000/3, about 6333. Interest can only add to this. – Ross Millikan Jul 9 '11 at 17:54

$10190.8$. First we need to bring a future value which is $19000$ at year $3$ to present(year zero) then subtract it from original value and then calculate the annual value for $3$ years with subtracted value.

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