Based on census data, the population of Freedonia is modeled by the function P(t)=265t^2+36600 people, where t represents the number of years after 1990. Use this function to determine in which year the population of Freedonia was increasing at a rate of 2650 people per year.
P ' (t) = 530t
530t = 2650
t = 5
looks like 1995
dP/dt = 530t
so dP/dt = 2650 when t = 5
So that makes it 1995
I got it Thank you!
To determine the year when the population of Freedonia was increasing at a rate of 2650 people per year, we need to find the derivative of the population function and set it equal to the given rate.
1. Start with the population function: P(t) = 265t^2 + 36600.
2. Take the derivative of P(t) with respect to t. The derivative of t^n is n*t^(n-1), so:
P'(t) = dP(t)/dt = d(265t^2 + 36600)/dt
= 2*265t (since the derivative of a constant is zero)
= 530t
3. Set the derivative equal to the given rate: 530t = 2650.
4. Divide both sides of the equation by 530 to solve for t: t = 5.
5. The value of t represents the number of years after 1990, so adding 5 years to 1990 gives us the year when the population was increasing at a rate of 2650 people per year: 1990 + 5 = 1995.
Therefore, in the year 1995, the population of Freedonia was increasing at a rate of 2650 people per year.