Question

the number of mosquitoes in brooklyn (in millions of mosquitoes) as a function of rainfall (in centimeters) is modeled by
m(x) = -x(x - 4)
what is the maximum possible number of mosquitoes?
million mosquitoes
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related content
quadratic word problems (factored form)

Explanation:

Step1: Rewrite the function

First, expand the function \( m(x)=-x(x - 4) \). Using the distributive property (FOIL method), we get \( m(x)=-x^{2}+4x \). This is a quadratic function in the form \( y = ax^{2}+bx + c \), where \( a=- 1 \), \( b = 4 \), and \( c = 0 \).

Step2: Find the x - coordinate of the vertex

For a quadratic function \( y=ax^{2}+bx + c \), the x - coordinate of the vertex (which gives the maximum or minimum value, and since \( a=-1<0 \), the parabola opens downwards, so the vertex is a maximum point) is given by the formula \( x=-\frac{b}{2a} \).
Substitute \( a=-1 \) and \( b = 4 \) into the formula: \( x=-\frac{4}{2\times(-1)}=-\frac{4}{-2} = 2 \).

Step3: Find the maximum value of the function

Now that we know the x - value at which the maximum occurs (\( x = 2 \)), we substitute \( x = 2 \) into the function \( m(x)=-x^{2}+4x \).
\( m(2)=-(2)^{2}+4\times2=-4 + 8=4 \).

Alternatively, we can use the factored form. The function \( m(x)=-x(x - 4) \) has roots at \( x = 0 \) and \( x = 4 \). For a quadratic function in factored form \( y=a(x - r_1)(x - r_2) \), the x - coordinate of the vertex is the mid - point of the roots. The mid - point of \( 0 \) and \( 4 \) is \( \frac{0 + 4}{2}=2 \), which is the same as we found earlier. Then substitute \( x = 2 \) into \( m(x) \): \( m(2)=-2(2 - 4)=-2\times(-2)=4 \).

Answer:

4