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A company is considering expanding their production capabilities with a new machine that costs $\mathrm{\$}56,000$ and has a projected lifespan of 9 years. They estimate the increased production will provide a constant $\mathrm{\$}7,000$ per year of additional income. Money can earn $1.4\mathrm{\%}$ per year, compounded continuously. Should the company buy the machine?

Yes, the present value of the machine is greater than the cost by $✓✓{\sigma }^{\mathrm{\infty }}\mathrm{\$}$ over the life of the machine

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Step 1 ：Calculate the exponent: $-0.014\ast 9=-0.126$

Step 2 ：Calculate the value of e raised to this power: $e^{-0.126}\approx 0.881$

Step 3 ：Substitute this back into the formula: $PV=7000\ast (1-0.881)/0.014$

Step 4 ：Calculate the term in the parentheses: $1-0.881=0.119$

Step 5 ：Multiply by the annual income: $7000\ast 0.119=833$

Step 6 ：Divide by the interest rate: $833/0.014\approx 59,500$

Step 7 ：The present value of the additional income that the machine will generate over its lifespan is approximately $59,500.

Step 8 ：Since the cost of the machine is $56,000 and the present value of the additional income is greater than the cost, the company should buy the machine.

Step 9 ：$\boxed{{\displaystyle \text{Therefore, the answer to the question 'Should the company buy the machine?' is yes.}}}$