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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)=−x2+14x m(x)=-x^2+14x What is the maximum possible number of mosquitoes? _____ \_\_\_\_\_ million 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: m(x)=−x2+14x m(x)=-x^2+14x What is the maximum possible number of mosquitoes? _____ \_\_\_\_\_ million mosquitoes
  1. Identify Function Type: Identify the type of function.\newlineThe function m(x)=−x2+14xm(x) = -x^2 + 14x is a quadratic function, which is a parabola. The coefficient of x2x^2 is negative, which means the parabola opens downwards.
  2. Find Parabola Vertex: Find the vertex of the parabola.\newlineThe vertex of a parabola given by the function f(x)=ax2+bx+cf(x) = ax^2 + bx + c is at the point (h,k)(h, k), where h=−b2ah = -\frac{b}{2a}. In this case, a=−1a = -1 and b=14b = 14.\newlineh=−b2a=−142×−1=142=7h = -\frac{b}{2a} = -\frac{14}{2 \times -1} = \frac{14}{2} = 7
  3. Calculate Maximum Value: Calculate the maximum value of the function.\newlineTo find the maximum value, which is the y-coordinate of the vertex kk, we substitute hh back into the function m(x)m(x).\newlinek=m(7)=−(7)2+14(7)=−49+98=49k = m(7) = -(7)^2 + 14(7) = -49 + 98 = 49
  4. Verify Result: Verify the result.\newlineSince the parabola opens downwards and we have found the vertex, the value of kk is indeed the maximum value of the function m(x)m(x).
  5. Answer Prompt: Answer the question prompt.\newlineThe maximum possible number of mosquitoes in millions as a function of rainfall in centimeters is 4949 million mosquitoes.

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