Question

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 temperature will result in the maximum number of fish?

Answer 1

• y=-2(x-9)²+200

Compare to vertex form of parabola y=a(x-h)²+k

Vertex:-

• (h,k)=(9,200)

As a is negative vertex is maximum

Max temperature=9°C

Answer 2

9 °C

Explanation:

Given function:

`P(x)=-2(x-9)^2+200`

The given function is a quadratic in vertex form.

Vertex form:
`y=a(x-h)^2+k` (where (h, k) is the vertex)

Therefore, the vertex is (9, 200)

The vertex is the minimum point for a parabola that opens upward.

The vertex is the maximum point for a parabola that opens downward.

The given function has a negative leading coefficient, therefore is opens downwards, and the vertex is the maximum point.

Therefore, the temperature (x-value) that will give the maximum number of fish (y-value) is the x-value of the vertex: 9 °C