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

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

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Step 1 ：Calculate the exponent: \(-0.014 * 9 = -0.126\)

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: \(PV = 7000 * (1 - 0.881) / 0.014\)

Step 4 ：Calculate the term in the parentheses: \(1 - 0.881 = 0.119\)

Step 5 ：Multiply by the annual income: \(7000 * 0.119 = 833\)

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 ：\(\boxed{\text{Therefore, the answer to the question 'Should the company buy the machine?' is yes.}}\)