Question

The number of gnats G(x), in millions, in a certain area depends on the May rainfall x, in inches. That relationship can be expressed by the quadratic function G(x) = - x2 + 18x How many inches of rainfall produces the maximum number of gnats?

Answer 1

9 inches of rainfall produces the maximum number of gnats.

Explanation:

Given :
`G(x) = - x^2 + 18x`

To Find :How many inches of rainfall produces the maximum number of gnats?

Solution:

`G(x) = - x^2 + 18x`

G(x) =The number of gnats in millions.

x denotes the length of rainfall

To Find the maximum length find the mid point of zeroes .Since parabolas are symmetrical, the x value of the max falls 1/2 way between the zeros.

`0= - x^2 + 18x`

`0= x^2-18x`

`0= x(x-18)`

So, x = 0 , x= 18

Now Formula of midpoint :
`x=(x_1+x_2)/(2)`

`x=(0+18)/(2)`

`x=9`

Hence 9 inches of rainfall produces the maximum number of gnats.