a population of foxes move into the river area by mr.radcliffe's house.at x months,the number of foxes p(x),can be modelled by the equation 4x*2+30x+105. a) what is the population of foxes at 7 months b)the average rate of change in the population from 0 to 7 months. c)the aproximate rate of change in the fox population at exactly 7 months. d)explain why the answers are different for a),b) and c).
To find the population of foxes at 7 months, we substitute x = 7 into the equation:
P(7) = 4(7)^2 + 30(7) + 105
= 4(49) + 210 + 105
= 196 + 210 + 105
= 511
So, the population of foxes at 7 months is 511.
To find the average rate of change in the population from 0 to 7 months, we calculate the difference in population and divide it by the difference in time:
Average rate of change = (P(7) - P(0)) / (7 - 0)
= (511 - P(0)) / 7
We do not have the equation for P(0) given, so we cannot calculate the exact average rate of change.
The approximate rate of change in the fox population at exactly 7 months can be found by calculating the derivative of the population function and evaluating it at x = 7. The derivative of the population function is:
P'(x) = 8x + 30
So, the approximate rate of change at 7 months is:
P'(7) = 8(7) + 30
= 56 + 30
= 86
Therefore, the approximate rate of change in the fox population at exactly 7 months is 86.
The answers for a), b), and c) are different because they represent different concepts.
a) asks for the population at a specific time point (7 months), b) asks for the average rate of change over a period of time (0 to 7 months), and c) asks for the approximate rate of change at a specific time point (7 months). Each question is looking at the population from a different perspective and therefore yields different answers.