need help with rates of change question (1 Viewer)

[kissyboy]

the number of ants in a colony is given by p = 12 + 11t - t^2, where P is the number of ants, in thousands, after t days from 1st of july.
1) find the rate of change of the population P with respect to time t.
2) find the rate of change of the population and the total population initially.
3) what is the maximum number of ants reached and in what date did this occured?
4) when did the colony died completely

 

[kurt.physics]

the number of ants in a colony is given by p = 12 + 11t - t^2, where P is the number of ants, in thousands, after t days from 1st of july.
1) find the rate of change of the population P with respect to time t.
2) find the rate of change of the population and the total population initially.
3) what is the maximum number of ants reached and in what date did this occured?
4) when did the colony died completely

1) The rate of change is the first derivative, so just find that by differentiating p.

2) initial means t = 0, that is, when the time is 0, ie the start.

3)for maximum, the first derivative is zero and the second derivative is negative. Or, as it is a quadratic, you could just find its axis of symmetry and hence its vertex.

4) this is when the population is zero, ie p = 0, ie 12 + 11t - t^2 = 0 (remember time cannot be negative)