Question
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?
centimeters
\[m(x)=-(x-5)^2+25\]
How many centimeters of rainfall will produce the maximum number of mosquitoes?
centimeters
Answers
GPT-4o mini
To determine the amount of rainfall that produces the maximum number of mosquitoes according to the given function \( m(x) = -(x-5)^2 + 25 \), we first recognize that this is a quadratic function in the standard form \( m(x) = a(x-h)^2 + k \), where \( (h, k) \) is the vertex of the parabola.
In this case:
- The coefficient \( a = -1 \) indicates that the parabola opens downwards, which means there is a maximum point.
- The expression \( (x-5)^2 \) suggests that the vertex (which provides \( h \)) is at \( x = 5 \).
Now, we know that \( h = 5 \) is the \( x \)-value at which the maximum occurs. The maximum number of mosquitoes will occur when the rainfall is:
\[
\boxed{5}
\]
centimeters.
In this case:
- The coefficient \( a = -1 \) indicates that the parabola opens downwards, which means there is a maximum point.
- The expression \( (x-5)^2 \) suggests that the vertex (which provides \( h \)) is at \( x = 5 \).
Now, we know that \( h = 5 \) is the \( x \)-value at which the maximum occurs. The maximum number of mosquitoes will occur when the rainfall is:
\[
\boxed{5}
\]
centimeters.