The revenue from a manufacturing process (in millions of dollars per year) is projected to follow the model $R = 400 + 0.07 t$ for 10 years. Over the same period of time, the cost (in millions of dollars per year) is projected to follow the model $C = 60 + 0.5 t ^ { 2 }$ , where t is the time (in years). Approximate the profit over the 10-year period, beginning with t = 0. Round your answer to two decimal places.
A) $\$ 3236.83$ million
B) $\$ 3378.50$ million
C) $\$ 3153.50$ million
D) $\$ 3235.67$ million
E) $\$ 3385.67$ million

Question

Quizplus · Accepted Answer

To approximate the profit over the 10-year period, we first calculate the total revenue and total cost over this period and then subtract the total cost from the total revenue.Total Revenue (R) from t=0 to t=10:$$R = \int_{0}^{10} (400 + 0.07t) dt$$$$= \left[400t + 0.035t^2\right]_{0}^{10}$$$$= (400*10 + 0.035*10^2) - (0)$$$$= 4000 + 3.5 = 4003.5$$Total Cost (C) from t=0 to t=10:$$C = \int_{0}^{10} (60 + 0.5t^2) dt$$$$= \left[60t + \frac{0.5t^3}{3}\right]_{0}^{10}$$$$= (60*10 + \frac{0.5*10^3}{3}) - (0)$$$$= 600 + \frac{500}{3} = 600 + 166.67 = 766.67$$Profit over the 10-year period:$$Profit = Total Revenue - Total Cost$$$$= 4003.5 - 766.67 = 3236.83$$Therefore, the approximate profit over the 10-year period is $\$3236.83$ million.

The Revenue from a Manufacturing Process (In Millions of Dollars R=400+0.07tR = 400 + 0.07 tR=400+0.07t

The Revenue from a Manufacturing Process (In Millions of Dollars $R = 400 + 0.07 t$