The population of a Canadian city is modelled by P(t)=12t^2+800t+40000, where t is the time in years. When t=0, the year is 2007. In what year is the poulation predicted to 300000?
To find the year when the population is predicted to be 300,000, we need to solve the equation:
12t^2 + 800t + 40000 = 300000
First, we subtract 300,000 from both sides of the equation:
12t^2 + 800t + 40000 - 300000 = 0
Simplifying:
12t^2 + 800t - 260000 = 0
Now, we can use the quadratic formula to solve for t:
t = (-b ± √(b^2 - 4ac)) / 2a
Substituting the values into the quadratic formula:
t = (-800 ± √(800^2 - 4 * 12 * -260000)) / (2 * 12)
Calculating:
t = (-800 ± √(640000 - (-1248000))) / 24
t = (-800 ± √(640000 + 1248000)) / 24
t = (-800 ± √1888000) / 24
Simplifying further:
t = (-800 ± 1374.77) / 24
Splitting into two equations:
t = (-800 + 1374.77) / 24 or t = (-800 - 1374.77) / 24
Solving each equation separately:
t1 = 574.77 / 24
t1 ≈ 23.95
t2 = -2174.77 / 24
t2 ≈ -90.62
Since time cannot be negative, we can discard the negative value of t.
Therefore, the time (t) is approximately 23.95 years from 2007.
To find the year, we add 23.95 years to 2007:
2007 + 23.95 ≈ 2031.95
Rounded to the nearest whole number, the year when the population is predicted to be 300,000 is 2032.
To find the year in which the population is predicted to be 300,000, we need to solve the equation P(t) = 300,000.
The given population model is P(t) = 12t^2 + 800t + 40,000.
Substituting P(t) with 300,000, the equation becomes:
300,000 = 12t^2 + 800t + 40,000.
Rearranging the equation to bring it to the form 12t^2 + 800t + 40,000 - 300,000 = 0:
12t^2 + 800t - 260,000 = 0.
Now we have a quadratic equation in the form at^2 + bt + c = 0, where:
a = 12
b = 800
c = -260,000
We can solve this quadratic equation using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a).
Substituting the given values:
t = (-800 ± √(800^2 - 4 * 12 * -260,000)) / (2 * 12).
Calculating the value inside the square root:
√(800^2 - 4 * 12 * -260,000) = √(640,000 - (-12,480,000)) = √(640,000 + 12,480,000) = √13,120,000 = 3,620.
Now substituting this value in the equation:
t = (-800 ± 3,620) / 24.
Using both the plus and minus signs:
t1 = (-800 + 3,620) / 24 = 2,820 / 24 = 117.5.
t2 = (-800 - 3,620) / 24 = -4,420 / 24 ≈ -184.2.
Since time cannot be negative, we disregard t2 = -184.2.
To find the year, we add the value of t to the initial year, 2007:
Year = 2007 + 117.5 = 2124.5.
Therefore, the population is predicted to reach 300,000 in the year 2124.
Man, you have the formula and the values. Just plug them in:
300000 = 12^t^2 + 800t + 40000
12t^2 + 800t - 260000 = 0
t = 117.5
year = 2124