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: $\newline$ $p(x)=-2 x^{2}+40 x-72$ $\newline$ Which temperatures will result in no fish (i.e. $0$ population)? $\newline$ Enter the lower temperature first. $\newline$ Lower temperature: $\square$ degrees Celsius $\newline$ Higher temperature: $\square$ degrees Celsius

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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:

\newline

p(x)=-2 x^{2}+40 x-72

\newline

Which temperatures will result in no fish (i.e.

0

population)?

\newline

Enter the lower temperature first.

\newline

Lower temperature:

\square

degrees Celsius

\newline

Higher temperature:

\square

degrees Celsius

Given quadratic function: We are given the quadratic function $p(x) = -2x^2 + 40x - 72$ , which models the fish population in thousands. We need to find the values of $x$ for which $p(x) = 0$ . This means we need to solve the quadratic equation $-2x^2 + 40x - 72 = 0$ .
Quadratic equation: To solve the quadratic equation, we can use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , where $a = -2$ , $b = 40$ , and $c = -72$ .
Calculating the discriminant: First, we calculate the discriminant, which is $b^2 - 4ac$ . Plugging in the values, we get discriminant: $\newline$ $= 40^2 - 4(-2)(-72)$ $\newline$ $= 1600 - 4(2)(72)$ $\newline$ $= 1600 - 576$ $\newline$ $= 1024$
Using the quadratic formula: Now we take the square root of the discriminant, $\sqrt{1024}$ , which is $32$ .
Calculating the values of x: We can now use the quadratic formula to find the two values of x. Plugging in the values, we get $x = \frac{{-40 \pm 32}}{{2 \times -2}}$
Solving for x: Calculating the two possible values for x, we get $x = \frac{{-40 + 32}}{{-4}}$ and $x = \frac{{-40 - 32}}{{-4}}$ .
Final result: Solving these, we get $x = \frac{-8}{-4} = 2$ and $x = \frac{-72}{-4} = 18$ .The two temperatures at which the fish population is zero are $2$ degrees Celsius and $18$ degrees Celsius. Since we need to enter the lower temperature first, the lower temperature is $2$ degrees Celsius and the higher temperature is $18$ degrees Celsius.