The quadratic P(x)=0.67x^2-0.049x+3 models the approximate population P(x), in thousands, for a species of fish......

Question

The quadratic P(x)=0.67x^2-0.049x+3 models the approximate population P(x), in thousands, for a species of fish in a local pond, x years after 1997. Durning what year will the population reach 26,826 fish?

Answer 1

Hi Brittany,

 

Let's begin by realizing that the number 26,826 will be represented as 26.826 in our quadratic function, since we are told the population, P(x), is measured in thousands.

 

Therefore, this is the equation which we will need to solve for x (# of years after 1997):

    y = 0.67x² - 0.049x + 3  [given quadratic equation]

    26.826 = 0.67x² - 0.049x + 3  [target y-value]

 

Now let's solve our equation for x:

    26.826 = 0.67x² - 0.049x + 3

    0.67x² - 0.049x + 3 - 26.826 = 0

    0.67x² - 0.049x - 23.826 = 0

    a = 0.67; b = -0.049; c = -23.826

    x = [-b +/- √(b² - 4ac)] / 2a

    x = {-(-0.049) +/- √[(-0.049)² - 4(0.67)(-23.826)]} / 2(0.67)

    x = [0.049 +/- √(0.002401 + 63.85368)] / 1.34

    x = (0.049 +/- √63.856081)/1.34

    x = (0.049 +/- 7.991) / 1.34

    x = (0.049 + 7.991)/1.34

    x = 8.046/1/34

    x = 6.0 (accept)

    x = (0.049 - 7.991) / 1.34

    x = -7.942/1.34

    x = -5.9 (reject)

 

We accept our positive answer (6 years) and add that to 1997 to get our final answer:

    1997 + 6 = 2003

                         (year population reaches 26,826)

 

Below is a link to our graph of this quadratic function:

 

https://dl.dropbox.com/s/ldsfm86eq1yncpu/Graph_of_Quadartic_Population_Function.png?raw=1

 

As can be seen on the graph, our answer is the time it takes to go from the y-intercept (y = -23.826) when x = 0 to the positive root of our equation (x = 6.0), which matches our algebraic solution above.

 

Thanks for submitting this problem and glad to help.

 

God bless, Jordan. 

Answer 2

Set P(x) equal to 26826.

 

 

26826 = 0.67x² - 0.049x + 3

 

 

Subtract 26826 on both sides of the equation.

 

0 = 0.67x² - 0.049x - 26823

 

 

Use the quadratic formula to solve for x:

 

 

x = (-b ± √(b² - 4ac)) / 2a

 

 

where:

a = 0.67

b = -0.049

c = -26823

 

 

Plug in these values into the formula to solve for x.  You will get two x values.  Choose the positive x value.  Then add the x value to 1997 to get the year.