(1 point) This problem is similar to one in your textbook.
Company QRS generated a net profit between 1992 and 1997 at a rate approximated by 27.5t+295 million dollars per year, where t is the time in years since 1992. Find the value, in 1992, of QRS's net profit over the 5-year period from 1992 to 1997. Assume that the interest rate is 2.25%, compounded continuously and round your answer to two decimal places.
Hmmm. I'm trying to pick between
5
∑(27.5t+295)*e^(.0225t) = 2026.95
t=1
∫[1,5] (27.5t+295)*e^(.0225t) dt = 1468.39
I'd be interested in seeing how you arrived at your answer. Maybe I'm wrong.
Or maybe it's
∫[0,5] (27.5t+295)*e^(.0225t) dt = 1931.82
To solve this problem, we need to find the integral of the given rate function, 27.5t + 295, over the interval from 1992 to 1997.
First, let's find the indefinite integral of the rate function:
∫(27.5t + 295) dt
Integrating 27.5t with respect to t gives us:
(27.5/2) t^2
Integrating 295 with respect to t gives us:
295t
Adding the two terms together gives the indefinite integral:
(27.5/2) t^2 + 295t + C
To evaluate this indefinite integral over the interval from 1992 to 1997, we need to find the definite integral by subtracting the value at the lower bound (1992) from the value at the upper bound (1997):
∫(27.5t + 295) dt [from 1992 to 1997]
= [(27.5/2) t^2 + 295t] [from 1992 to 1997]
= [(27.5/2) (1997)^2 + 295(1997)] - [(27.5/2) (1992)^2 + 295(1992)]
Now we can substitute the values and calculate the net profit over the 5-year period from 1992 to 1997.