a population of foxes move into the river area Mr.Radcliffe's house.At x months,the number of foxes P(x),can be modelled by the equation P(x)=4x*2+30x+105.Find the : a) population of foxes at 7 months. b)the average rate of change in the population from 0 tp 7 motnhs. c)the approximate rate of change in the fox population at exactlty 7 months. d)explain why the answers are different for a,b,c

Answer 1

a) To find the population of foxes at 7 months, we plug in x = 7 into the equation P(x):

P(7) = 4(7)^2 + 30(7) + 105
P(7) = 4(49) + 210 + 105
P(7) = 196 + 210 + 105
P(7) = 511

Therefore, the population of foxes at 7 months is 511.

b) To find the average rate of change in the population from 0 to 7 months, we first find the population at 0 months and 7 months:

P(0) = 4(0)^2 + 30(0) + 105
P(0) = 0 + 0 + 105
P(0) = 105

The population at 0 months is 105.

The average rate of change is given by:

Average Rate of change = (P(7) - P(0))/(7-0)
Average Rate of change = (511 - 105)/7
Average Rate of change = 406/7
Average Rate of change ≈ 58

c) The approximate rate of change in the fox population at exactly 7 months can be found by taking the derivative of the population function with respect to x and evaluating it at x = 7:

P'(x) = 8x + 30

P'(7) = 8(7) + 30
P'(7) = 56 + 30
P'(7) = 86

Therefore, the approximate rate of change in the fox population at exactly 7 months is 86.

d) The answers are different for parts a, b, and c because:
- Part a asks for the actual population of foxes at 7 months, which is determined by plugging in x = 7 into the equation.
- Part b asks for the average rate of change in the population over the interval from 0 to 7 months, which is calculated by finding the difference in population between 7 months and 0 months divided by the time interval.
- Part c asks for the approximate rate of change at exactly 7 months, which is determined by taking the derivative of the population function with respect to x and evaluating it at x = 7.