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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)+14 x What is the maximum possible number of mosquitoes? million mosquitoes

The number of mosquitoes in Anchorage, Alaska (in millions of mosquitoes) as a function of rainfall (in centimeters) is modeled by:\newlinem(x)=−x2+14xm(x)=-x^{2}+14x\newlineWhat is the maximum possible number of mosquitoes?\newlinemillion mosquitoes

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Q. The number of mosquitoes in Anchorage, Alaska (in millions of mosquitoes) as a function of rainfall (in centimeters) is modeled by:\newlinem(x)=−x2+14xm(x)=-x^{2}+14x\newlineWhat is the maximum possible number of mosquitoes?\newlinemillion mosquitoes
  1. Identify Quadratic Function: The function given is m(x)=−x2+14xm(x) = -x^2 + 14x. This is a quadratic function in the form of f(x)=ax2+bx+cf(x) = ax^2 + bx + c, where a=−1a = -1, b=14b = 14, and c=0c = 0. Since the coefficient of x2x^2 is negative (a=−1a = -1), the parabola opens downwards, which means it has a maximum point.
  2. Find Vertex Formula: To find the maximum value of the function, we need to find the vertex of the parabola. The xx-coordinate of the vertex of a parabola given by f(x)=ax2+bx+cf(x) = ax^2 + bx + c is found using the formula −b2a-\frac{b}{2a}.
  3. Calculate x-coordinate of Vertex: Substitute a=−1a = -1 and b=14b = 14 into the formula to find the x-coordinate of the vertex: x=−b2a=−142∗(−1)=−14−2=7x = -\frac{b}{2a} = -\frac{14}{2*(-1)} = -\frac{14}{-2} = 7.
  4. Substitute xx into Function: Now that we have the xx-coordinate of the vertex, we can find the maximum number of mosquitoes by substituting x=7x = 7 into the function m(x)m(x): m(7)=−(7)2+14×7m(7) = -(7)^2 + 14 \times 7.
  5. Calculate Maximum Mosquitoes: Calculate the value of m(7)m(7): m(7)=−(49)+98=49m(7) = -(49) + 98 = 49 million mosquitoes.
  6. Conclusion: The maximum possible number of mosquitoes is 4949 million mosquitoes.

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