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)=-2x^2+40x-72

1) What type of function is being shown?

2) What other characteristics of this type of function?

3) Determine which time will result in no fish, a population of zero.

4) use a different strategy to determine which time will result in no fish.

5) Did both strategies give you the same answer? Should they?

6) Determine which temperature will result in the largest population of fish. Explain how you determine this.

Answer 1

1) The function being shown is a quadratic function.

2) Characteristics of a quadratic function include having a degree of 2, a graph that forms a parabola, and an equation in the form of ax^2 + bx + c.

3) To determine when the population will be zero, we can set p(x) equal to zero and solve for x:
-2x^2 + 40x - 72 = 0

4) Another strategy to determine when the population will be zero is by factoring the quadratic equation, if possible. If factoring is not feasible, we can use the quadratic formula.

5) Both strategies should give the same answer when solving for the time resulting in a population of zero. However, the approach may differ in terms of complexity or computational ease.

6) To determine the temperature that will result in the largest population of fish, we need to find the vertex of the quadratic function. The x-coordinate of the vertex represents the temperature at which the fish population is maximized. In this case, the vertex can be found using the formula x = -b / (2a), where a, b, and c are the coefficients of the quadratic equation.

Answer 2

3) To determine the time at which the fish population is zero:

We have the quadratic equation: -2x^2 + 40x - 72 = 0

Using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a)

Substituting the values from our equation: a = -2, b = 40, c = -72

x = (-40 ± √(40^2 - 4(-2)(-72))) / (2(-2))

Simplifying further:

x = (-40 ± √(1600 - 576)) / (-4)

x = (-40 ± √(1024)) / (-4)

x = (-40 ± 32) / (-4)

So, the solutions for x (temperature) that result in a population of zero are:

x1 = (-40 + 32) / (-4) = -8 / (-4) = 2

x2 = (-40 - 32) / (-4) = -72 / (-4) = 18

Therefore, the fish population will be zero at temperature x = 2°C and x = 18°C.

6) To determine the temperature that results in the largest population of fish (maximum point):

The x-coordinate of the vertex can be found using the formula: x = -b / (2a)

In our equation, a = -2 and b = 40:

x = -40 / (2(-2)) = -40 / (-4) = 10

So, the temperature x = 10°C will result in the largest population of fish. The y-coordinate of the vertex represents the maximum population.

1) The given function is a quadratic function.

2) Characteristics of a quadratic function include:

- It is a polynomial function of degree 2.

- The graph of a quadratic function is a parabola.

- It has a vertex, which is either a minimum or maximum point, depending on the coefficient of the leading term.

- The graph is symmetric about the vertical line passing through the vertex.

- The function can have either a positive or negative leading coefficient, which determines the concavity of the parabola.

3) To determine the time at which the fish population is zero, we need to find the value of x (temperature) that makes the function p(x) equal to zero:

-2x^2 + 40x - 72 = 0

To solve this quadratic equation, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = -2, b = 40, and c = -72. Plugging in these values into the quadratic formula, we can find the values of x that result in a population of zero.

4) An alternative strategy to determine when the fish population is zero is by factoring the quadratic equation if possible. However, the given quadratic equation doesn't appear to be easily factorable, so using the quadratic formula is a more suitable approach.

5) Both strategies should give the same answer. Whether using the quadratic formula or factoring, the solutions for x (temperature) that result in a population of zero should be identical. The quadratic formula is a general method that works for all quadratic equations, even when factoring is not immediately apparent.

6) To determine the temperature that results in the largest population of fish, we need to find the vertex of the quadratic function. The x-coordinate of the vertex can be found using the formula:

x = -b / (2a)

In this case, a = -2 and b = 40. Plugging in these values, we can calculate the temperature (x) at which the fish population is maximized. The y-coordinate of the vertex will represent the largest population of fish.

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