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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?

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Answer:

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)

User Trevor Pilley
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