Homework SourseThe total accumulated costs Ct and revenues Rt in thousands
The total accumulated costs C(t) and revenues R(t) (in thousands of dollars), respectively, for a photocopying machine satisfy
C\'(t)=((1/35)(t^2)) and R\'(t)= ((7t^2)(e^(-t^3))) where t is the time in years.
Find the useful life of the machine, to the nearest year.
Using the useful life of the machine rounded to the nearest year, the total profit accumulated during the useful life of the machine is $_____.
C\'(t)=((1/35)(t^2)) and R\'(t)= ((7t^2)(e^(-t^3))) where t is the time in years.
Find the useful life of the machine, to the nearest year.
Using the useful life of the machine rounded to the nearest year, the total profit accumulated during the useful life of the machine is $_____.
Solution
Useful life of a machine: R(t) >= C(t)
R(t) = (7/3)(1 - e^(-t^3))
C(t) = (t^3/105)
(7/3)(1 - e^(-t^3)) >= (t^3/105)
t^3/(1 - e^(-t^3)) <= 245
t = 6 yrs (nearly)
Profit accumulated = R(6) - C(6) = 7/3 - 216/105 = 0.2762 (thousands of dollars)
