The machine earns the company revenue at a continuous rate of 62000 t + 38000 dollars per year during the first six months of operation, and at the continuous rate of $69000 per year after the first six months. The cost of the machine is $170000. The interest rate is 5% 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.
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.
To find the present value of the revenue earned by the machine during the first year of operation, we need to calculate the integral of the revenue function over the interval of 0 to 1 (representing one year).
The revenue function is given by:
R(t) = 62000t + 38000, for 0 ≤ t ≤ 0.5
R(t) = 69000, for 0.5 < t ≤ 1
Therefore, the total revenue earned within the first year can be calculated as follows:
∫[0,0.5] (62000t + 38000) dt + ∫[0.5,1] 69000 dt
Integrating the first part of the revenue function:
∫[0,0.5] (62000t + 38000) dt = [31000t^2 + 38000t] evaluated from 0 to 0.5
= (31000(0.5)^2 + 38000(0.5)) - (31000(0)^2 + 38000(0))
= (31000(0.25) + 19000)
Integrating the second part of the revenue function:
∫[0.5,1] 69000 dt = 69000(t) evaluated from 0.5 to 1
= 69000(1) - 69000(0.5)
= 69000 - 34500
Adding both calculations:
(31000(0.25) + 19000) + (69000 - 34500)
Simplifying further:
7750 + 34500
The total present value of the revenue earned by the machine during the first year of operation is $42250. Round your answer to the nearest cent.
To determine how long it will take for the machine to pay for itself, we need to find the time (t) when the present value of the revenue equals the cost of the machine.
Let's set up an equation:
PV = 170000
Using the revenue function for the first year:
PV = ∫[0,t] (62000t + 38000) dt + ∫[t,1] 69000 dt
Simplifying and integrating:
170000 = [31000t^2 + 38000t] evaluated from 0 to t + 69000(t - t)
170000 = (31000t^2 + 38000t) + 69000(t - t)
Simplifying further:
170000 = 31000t^2 + 38000t
Rearranging the equation to solve for t:
31000t^2 + 38000t - 170000 = 0
We can now use the quadratic formula to find the values of t. The quadratic formula states that for an equation in the form of ax^2 + bx + c = 0, the solutions are given by:
t = (-b ± sqrt(b^2 - 4ac)) / (2a)
Using the values from our equation:
a = 31000, b = 38000, and c = -170000
Calculating the discriminant:
b^2 - 4ac = (38000)^2 - 4(31000)(-170000)
Simplifying:
1444000000 + 21760000000 = 23204000000
Taking the square root of the discriminant:
sqrt(23204000000) ≈ 152305.77
Using the quadratic formula:
t = (-38000 ± 152305.77) / (2 * 31000)
Calculating both values of t:
t1 = (-38000 + 152305.77) / (2 * 31000)
t2 = (-38000 - 152305.77) / (2 * 31000)
t1 ≈ 1.349
t2 ≈ -1.672
Since time cannot be negative, we only consider the positive value of t: t ≈ 1.349
Therefore, it will take approximately 1.35 years for the machine to pay for itself. Round your answer to the nearest hundredth.