using the following information in place of that in the book. The machine earns the company revenue at a continuous rate of 50000 t + 35000 dollars per year during the first six months of operation, and at the continuous rate of $60000 per year after the first six months. The cost of the machine is $175000. The interest rate is 7% per year, compounded continuously.
a) Find the present value of the revenue earned by the machine during the first year of operation. Round your answer to the nearest cent.
Value: $
b) Determine how long it will take for the machine to pay for itself; that is, how long until the present value of the revenue is equal to the cost of the machine. Round your answer to the nearest hundredth.
Years:
a) To find the present value of the revenue earned by the machine during the first year of operation, we need to calculate the present value of both revenue streams.
During the first six months, the revenue earned is given by the equation:
R1(t) = (50000t + 35000)
Where t represents time in years. Since we're considering the first year, t can take values from 0 to 1/2.
To find the present value of this revenue stream, we use the formula for continuous compounding:
PV1 = ∫[0 to 1/2] (50000t + 35000)e^(-0.07t) dt
Integrating this equation gives us:
PV1 = [50000t^2/2 - 35000e^(-0.07t)] evaluated from t = 0 to t = 1/2
Simplifying it further:
PV1 = (50000(1/2)^2/2 - 35000e^(-0.07(1/2))) - (50000(0)^2/2 - 35000e^(-0.07(0)))
PV1 = (12500 - 35000e^(-0.035)) - (0 - 35000e^(0))
PV1 = 12500 - 35000e^(-0.035)
Now let's calculate the present value of the revenue earned during the second six months, which is a constant $60000 per year. Using the same continuous compounding formula:
PV2 = ∫[1/2 to 1] 60000e^(-0.07t) dt
Integrating this equation gives us:
PV2 = [-60000e^(-0.07t)] evaluated from t = 1/2 to t = 1
Simplifying it further:
PV2 = -60000e^(-0.07(1)) - (-60000e^(-0.07(1/2)))
PV2 = -60000e^(-0.07) + 60000e^(-0.035)
Now, we can find the total present value of the revenue earned during the first year by adding PV1 and PV2:
Total Present Value = PV1 + PV2
Total Present Value = (12500 - 35000e^(-0.035)) + (-60000e^(-0.07) + 60000e^(-0.035))
Now, we can substitute the values into a calculator and round the answer to the nearest cent.
b) To determine how long it will take for the machine to pay for itself, we need to find the point at which the present value of the revenue is equal to the cost of the machine, which is $175000. We can set up the equation:
PV1 + PV2 = 175000
Substituting the expressions for PV1 and PV2 that we derived in part a:
(12500 - 35000e^(-0.035)) + (-60000e^(-0.07) + 60000e^(-0.035)) = 175000
Now, we need to solve this equation for t, which represents the time it takes for the machine to pay for itself. We can do this by using numerical approximation methods such as the Newton-Raphson method or by using a calculator or software that can solve equations numerically.
Once we have obtained the value of t, we can round it to the nearest hundredth to find how long it will take for the machine to pay for itself.