Question

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the number of mosquitoes in minneapolis, minnesota (in millions of mosquitoes) as a function of rainfall (in centimeters) is modeled by:
$m(x) = -(x - 5)^2 + 25$
how many centimeters of rainfall will produce the maximum number of mosquitoes?
\boxed{} centimeters
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Explanation:

Step1: Identify the form of the function

The function \( m(x) = -(x - 5)^2 + 25 \) is in vertex form of a quadratic function, which is \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex of the parabola.

Step2: Determine the direction of the parabola

In the given function, \( a = -1 \), which is less than 0. So, the parabola opens downward, and the vertex represents the maximum point of the function.

Step3: Find the x - coordinate of the vertex

For the vertex form \( y = a(x - h)^2 + k \), the x - coordinate of the vertex is \( h \). In the function \( m(x)=-(x - 5)^2+25 \), comparing with \( y = a(x - h)^2 + k \), we have \( h = 5 \). This \( x \) - value (where \( x \) represents the rainfall in centimeters) gives the maximum value of the function \( m(x) \) (the number of mosquitoes).

Answer:

5