Final answer:
To calculate the time for the machinery to pay for itself, use the present value formula for continuous compounding, solve for time, and then plug in the values to find the number of years, rounding to two decimal places.
Step-by-step explanation:
To determine how long it takes for the machinery to pay for itself, we calculate the present value of the profit generated and set it equal to the initial cost of the machinery. The present value (PV) of the profit that the machinery generates can be determined using the formula for continuous compounding: PV = P × e^{-rt}, where P is the profit generated per year, r is the interest rate, and t is time in years. To find the break-even time, i.e., when the machinery has paid for itself, we solve the equation 140,000 = 90,000 × e^{-0.075t}.
To isolate t, we divide both sides by 90,000, getting ⅛ e^{-0.075t}, and then take the natural logarithm of both sides which gives us ln(⅛) = -0.075t. Solving for t gives us t = -⅛/0.075.
Lastly, we plug the values into the equation to get the time in years and round our answer to two decimal places.