the population of a town is modelled by the functions p(t)=6t2+110t+3000. when will the population reach 6000? thanks
solve the quadratic
6t^2 + 110t + 3000 = 6000
3t^2 + 55t - 1500 = 0
(t-15)(3t + 100) = 0
t = 15 or t is a negative, which would make no sense
it will take 15 years
To find when the population will reach 6000, we need to solve the equation p(t) = 6000, where p(t) is the population function.
Given: p(t) = 6t^2 + 110t + 3000
Step 1: Set p(t) equal to 6000:
6t^2 + 110t + 3000 = 6000
Step 2: Rearrange the equation and set it to zero by subtracting 6000 from both sides:
6t^2 + 110t + 3000 - 6000 = 0
Simplifying:
6t^2 + 110t - 3000 = 0
Step 3: Now, we need to solve this quadratic equation. There are different methods to solve quadratic equations, such as factoring, completing the square, or using the quadratic formula.
In this case, let's use the quadratic formula:
t = (-b ± sqrt(b^2 - 4ac)) / (2a)
For our equation:
a = 6
b = 110
c = -3000
Plugging in the values into the quadratic formula:
t = (-110 ± sqrt(110^2 - 4(6)(-3000))) / (2(6))
Simplifying further:
t = (-110 ± sqrt(12100 + 72000)) / 12
t = (-110 ± sqrt(84100)) / 12
t = (-110 ± 290) / 12
Step 4: We have two possible solutions here. Let's calculate both:
t1 = (-110 + 290) / 12 = 180 / 12 = 15
t2 = (-110 - 290) / 12 = -400 / 12 ≈ -33.33 (which is not a valid solution since time cannot be negative)
So, the population will reach 6000 after approximately 15 units of time.
Therefore, the population will reach 6000 in 15 units of time.