Question

A government agency estimates the number of young adults (ages 18 to 24) in a particular country to be 31,000 (in thousands) in 2010 and changing at the rate of −x2 + 90x − 200 thousand per year, where x is the number of years since 2010. Find a formula for the size of this population at any time x. [Hint: Keep all calculations in units of thousands.]

Answer 1

`\mathbf{L(x)= ( - (1)/(3))x^3 + 45x^2 -200x +31000}`

Explanation:

From the given information:

Let assume the population is denoted by L

The rate of change of the young adults per year given can be represented as;

`(dL)/(dx)= -x^2 +90x - 200`

where;

x = 0 since 2010

`dL = -x^2 +90x -200 dx`

`L = \int( -x^2 +90x -200 ) \ dx`

`L = - (1)/(3)x^3 + 45x^2 -200x +C`

here;

L(0) = 31000

∴

`- (1)/(3)(0)^3 + 45(0)^2 -200(0)+C= 31000`

C = 31000

`\mathbf{L(x)= ( - (1)/(3))x^3 + 45x^2 -200x +31000}`