Solve the problem.
-After a new firm starts in business, it finds that its rate of profits (in hundreds of dollars per year) after t years of operation is given by $\frac { \mathrm { dP } } { \mathrm { dt } } = 3 \mathrm { t } ^ { 2 } + 2 \mathrm { t } + 6$. Find the profit in year 7 of the operation.
A) $14,600
B) $36,100
C) $42,000
D) $28,800

Question

Quizplus · Accepted Answer

Explanation:The profit in year 7 is given by the integral of the rate of profits from year 0 to year 7:$$\int_0^7 (3t^2 + 2t + 6) dt = [t^3 + t^2 + 6t]_0^7 = 7^3 + 7^2 + 6(7) - 0^3 - 0^2 - 6(0) = 343 + 49 + 42 = 434$$Therefore, the profit in year 7 is $43400.

Solve the Problem dPdt=3t2+2t+6\frac { \mathrm { dP } } { \mathrm { dt } } = 3 \mathrm { t } ^ { 2 } + 2 \mathrm { t } + 6dtdP​=3t2+2t+6

Solve the Problem $\frac { \mathrm { dP } } { \mathrm { dt } } = 3 \mathrm { t } ^ { 2 } + 2 \mathrm { t } + 6$