Question

The number of mosquitoes in Anchorage, Alaska (in millions of mosquitoes) as a function of rainfall (in centimeters) is modeled by: m(x)=-x^2+14x How many centimeters of rain will produce the maximum number of mosquitoes?

Answer 1

7 centimeters of rain will produce the maximum number of mosquitoes (49 millions of mosquitoes)

Explanation:

we have the function

`m(x)=-x^2+14x`

where

m(x) is the number of mosquitoes in Anchorage, Alaska (in millions of mosquitoes)

x is the rainfall (in centimeters)

This is a vertical parabola open downward

The vertex represent a maximum

The x-coordinate of the vertex represent the centimeters of rain that will produce the maximum number of mosquitoes

so

Convert the quadratic equation into vertex form

`m(x)=-x^2+14x`

Factor -1

`m(x)=-(x^2-14x)`

Complete the square

`m(x)=-(x^2-14x+7^2)+7^2`

`m(x)=-(x^2-14x+49)+49`

Rewrite as perfect squares

`m(x)=-(x-7)^2+49`

The vertex is the point (7,49)

therefore

7 centimeters of rain will produce the maximum number of mosquitoes (49 millions of mosquitoes)