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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? centimeters

The number of mosquitoes in Minneapolis, Minnesota (in millions of mosquitoes) as a function of rainfall (in centimeters) is modeled by:\newlinem(x)=−(x−5)2+25m(x) = -(x - 5)^{2} + 25\newlineHow many centimeters of rainfall will produce the maximum number of mosquitoes?\newlinecentimeters\text{centimeters}

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Q. The number of mosquitoes in Minneapolis, Minnesota (in millions of mosquitoes) as a function of rainfall (in centimeters) is modeled by:\newlinem(x)=−(x−5)2+25m(x) = -(x - 5)^{2} + 25\newlineHow many centimeters of rainfall will produce the maximum number of mosquitoes?\newlinecentimeters\text{centimeters}
  1. Identify Function Type: Identify the type of function given for the number of mosquitoes as a function of rainfall.\newlineThe function m(x)=−(x−5)2+25m(x) = -(x-5)^2 + 25 is a quadratic function in the form of m(x)=−a(x−h)2+km(x) = -a(x-h)^2 + k, where (h,k)(h, k) is the vertex of the parabola.
  2. Determine Parabola Vertex: Determine the vertex of the parabola.\newlineSince the quadratic function is in vertex form, the vertex (h,k)(h, k) can be directly read from the function as (5,25)(5, 25).
  3. Understand Vertex Role: Understand the role of the vertex in a quadratic function. The vertex represents the maximum or minimum point of the parabola. Since the coefficient of the squared term is negative (−1-1), the parabola opens downwards, and thus the vertex is the maximum point.
  4. Identify Maximum Rainfall: Identify the rainfall that corresponds to the maximum number of mosquitoes. The xx-coordinate of the vertex, hh, represents the amount of rainfall that will produce the maximum number of mosquitoes. From the vertex (5,25)(5, 25), we can see that h=5h = 5.
  5. Conclude Final Answer: Conclude the solution with the final answer. The maximum number of mosquitoes will be produced when there are 55 centimeters of rainfall.

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