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The fish population in a certain part of the ocean (in thousands of fish) as a function of the water's temperature (in degrees Celsius) is modeled by: P(x)=-2(x-9)^(2)+200 What is the maximum number of fish? thousand fish

The fish population in a certain part of the ocean (in thousands of fish) as a function of the water's temperature (in degrees Celsius) is modeled by:\newlineP(x)=−2(x−9)2+200 P(x)=-2(x-9)^{2}+200 \newlineWhat is the maximum number of fish?\newlinethousand fish

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Q. The fish population in a certain part of the ocean (in thousands of fish) as a function of the water's temperature (in degrees Celsius) is modeled by:\newlineP(x)=−2(x−9)2+200 P(x)=-2(x-9)^{2}+200 \newlineWhat is the maximum number of fish?\newlinethousand fish
  1. Given quadratic function: We are given the quadratic function P(x)=−2(x−9)2+200P(x) = -2(x - 9)^2 + 200, which models the fish population in thousands. To find the maximum number of fish, we need to find the vertex of this parabola. Since the coefficient of the squared term is negative −2-2, the parabola opens downwards, and the vertex will give us the maximum value.
  2. Finding the vertex: The vertex form of a parabola is given by P(x)=a(x−h)2+kP(x) = a(x - h)^2 + k, where (h,k)(h, k) is the vertex of the parabola. In our function, h=9h = 9 and k=200k = 200. This means the vertex is at the point (9,200)(9, 200).
  3. Interpreting the vertex: The xx-coordinate of the vertex, hh, represents the temperature at which the maximum number of fish is found, and the yy-coordinate, kk, represents the maximum number of fish in thousands. Since we are only interested in the maximum number of fish, we look at the yy-coordinate, which is 200200.
  4. Maximum number of fish: Therefore, the maximum number of fish, in thousands, is 200200.

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