A study indicates that x months from now the population of a certain city will be increasing at the rate of $( 3 + 3 x ) x ^ { - 1 / 2 }$ people per month.By how much will the population increase over the next 9 months?
A)54 people
B)62 people
C)72 people
D)82 people

Question

Quizplus · Accepted Answer

To find the total increase in population over the next 9 months, we need to integrate the given rate of increase, $(3 + 3x)x^{-1/2}$, from 0 to 9 months. The integral of $(3 + 3x)x^{-1/2}$ with respect to $x$ gives us the total increase in population over the specified period.$$\int_0^9 (3 + 3x)x^{-1/2} dx = \int_0^9 3x^{-1/2} dx + \int_0^9 3x^{1/2} dx$$Solving the integrals:$$= 6x^{1/2} \Big|_0^9 + 2x^{3/2} \Big|_0^9$$$$= 6(9)^{1/2} - 6(0)^{1/2} + 2(9)^{3/2} - 2(0)^{3/2}$$$$= 6(3) + 2(27)$$$$= 18 + 54 = 72$$Therefore, the population will increase by 72 people over the next 9 months.

A Study Indicates That X Months from Now the Population (3+3x)x−1/2( 3 + 3 x ) x ^ { - 1 / 2 }(3+3x)x−1/2

A Study Indicates That X Months from Now the Population $( 3 + 3 x ) x ^ { - 1 / 2 }$