Answer:
It will take approximately 68.2 years for the Caribou population to reach 21,000.
Step-by-step explanation:
The population P of Caribou in the Yukon is modeled by P=6x²-120x+1250 where x is the number of years from today.
If the population P of Caribou in the Yukon is 21000, then, replacing you get:
21000= 6x²-120x+1250
So:
0= 6x²-120x+1250 -21000
0= 6x²-120x - 19750
This is a quadratic function of the form: 0= ax²+ bx +c, where: a=6, b= -120 and c=-19750
To solve a quadratic function and determine the value of x you can use the expression:
[tex]x1,x2=\frac{-b+-\sqrt{b^{2}-4*a*c } }{2*a}[/tex]
Replacing the values of a, b and c and solving you get:
[tex]x1,x2=\frac{-(-120)+-\sqrt{(-120)^{2}-4*6*(-19750) } }{2*6}[/tex]
[tex]x1,x2=\frac{120+-\sqrt{14400+474000 } }{12}[/tex]
[tex]x1,x2=\frac{120+-\sqrt{488400 } }{12}[/tex]
[tex]x1,x2=\frac{120+-698.86 }{12}[/tex]
So:
[tex]x1=\frac{120+698.86 }{12}[/tex]
x1= 68.24 ≅ 68.2
and
[tex]x2=\frac{120-698.86 }{12}[/tex]
x2= -48. 24 ≅ -48.2
As time cannot be negative, it will take approximately 68.2 years for the Caribou population to reach 21,000.
