The population of a planned seaside community in Florida is given by the function P(t)=3000+200t+0.1t^2, where t represents the number of years since the community was incorporated in 1985. 10 years represents the population in 1995 and 20 represents the population 2005.
find P(10) and P(20). What do these values represents?
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find the average rate of change of P between t=10 and t=20. What does the number represent?
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Approximately when will the population be 6000?
To find P(10) and P(20), we plug in the values of t into the function:
P(10) = 3000 + 200(10) + 0.1(10)^2
P(10) = 3000 + 2000 + 1
P(10) = 5001
So, P(10) represents the population in 1995, which is 5001.
P(20) = 3000 + 200(20) + 0.1(20)^2
P(20) = 3000 + 4000 + 4
P(20) = 7004
So, P(20) represents the population in 2005, which is 7004.
The average rate of change of P between t=10 and t=20 can be found using the formula:
Average rate of change = (P(20) - P(10)) / (20 - 10)
Average rate of change = (7004 - 5001) / 10
Average rate of change = 200.3
The number 200.3 represents the average rate of change of the population per year between 1995 and 2005.
To find approximately when the population will be 6000, we set P(t) = 6000 and solve for t:
6000 = 3000 + 200t + 0.1t^2
0.1t^2 + 200t - 3000 = 0
Using a quadratic formula, we get t ≈ 11.13 years
So, approximately when the population will be 6000 is around 11.13 years after 1985, which is in 1996.13.