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

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)=−2x2+40x−72 p(x)=-2 x^{2}+40 x-72 \newlineWhich temperatures will result in no fish (i.e. 00 population)?\newlineEnter the lower temperature first.\newlineLower temperature: □\square degrees Celsius\newlineHigher 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:\newlinep(x)=−2x2+40x−72 p(x)=-2 x^{2}+40 x-72 \newlineWhich temperatures will result in no fish (i.e. 00 population)?\newlineEnter the lower temperature first.\newlineLower temperature: □\square degrees Celsius\newlineHigher temperature: □\square degrees Celsius
  1. Given quadratic function: We are given the quadratic function p(x)=−2x2+40x−72p(x) = -2x^2 + 40x - 72, which models the fish population in thousands. We need to find the values of xx for which p(x)=0p(x) = 0. This means we need to solve the quadratic equation −2x2+40x−72=0-2x^2 + 40x - 72 = 0.
  2. Quadratic equation: To solve the quadratic equation, we can use the quadratic formula x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, where a=−2a = -2, b=40b = 40, and c=−72c = -72.
  3. Calculating the discriminant: First, we calculate the discriminant, which is b2−4acb^2 - 4ac. Plugging in the values, we get discriminant:\newline=402−4(−2)(−72)= 40^2 - 4(-2)(-72)\newline=1600−4(2)(72)= 1600 - 4(2)(72)\newline=1600−576= 1600 - 576\newline=1024= 1024
  4. Using the quadratic formula: Now we take the square root of the discriminant, 1024\sqrt{1024}, which is 3232.
  5. 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=−40±322×−2x = \frac{{-40 \pm 32}}{{2 \times -2}}
  6. Solving for x: Calculating the two possible values for x, we get x=−40+32−4x = \frac{{-40 + 32}}{{-4}} and x=−40−32−4x = \frac{{-40 - 32}}{{-4}}.
  7. Final result: Solving these, we get x=−8−4=2x = \frac{-8}{-4} = 2 and x=−72−4=18x = \frac{-72}{-4} = 18.The two temperatures at which the fish population is zero are 22 degrees Celsius and 1818 degrees Celsius. Since we need to enter the lower temperature first, the lower temperature is 22 degrees Celsius and the higher temperature is 1818 degrees Celsius.

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