Question

the number of mosquitoes in anchorage, alaska (in millions of mosquitoes) as a function of rainfall (in centimeters) is modeled by:
$m(x) = -x^2 + 14x$
what is the maximum possible number of mosquitoes?
\boxed{} million mosquitoes

Explanation:

Step1: Identify the function type

The function \( m(x) = -x^2 + 14x \) is a quadratic function in the form \( ax^2 + bx + c \) (here \( c = 0 \)), and since \( a=-1<0 \), the parabola opens downward, so the vertex is the maximum point.

Step2: Find the x-coordinate of the vertex

For a quadratic function \( ax^2 + bx + c \), the x-coordinate of the vertex is given by \( x = -\frac{b}{2a} \). Here, \( a = -1 \) and \( b = 14 \), so \( x = -\frac{14}{2\times(-1)}=\frac{-14}{-2} = 7 \).

Step3: Find the maximum value

Substitute \( x = 7 \) into the function \( m(x) \): \( m(7)=-(7)^2 + 14\times7=-49 + 98 = 49 \).

Answer:

49