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

Answer:

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

Step-by-step 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)