Question

The observed rabbit population on an island is given by the function p(t) =-4t^2+80t+1200, where t is the time, in years, since the teachers began observing the rabbits. According to this quadratic function, after how many years will the rabbit population reach its peak

Answer 1

The rabbit population will reach its peak after 10 years.

Explanation:

Suppose we have a quadratic function in the following format:

`p(t) = at^(2) + bt + c`

The vertex of the function is the point:

`(t_(v), p(t_(v))`

In which

`t_(v) = -(b)/(2a)`

If a is negative, the vertex is a peak.

In this question:

`p(t) = -4t^(2) + 80t + 1200`

So

`a = -4, b = 80, c = 1200`

According to this quadratic function, after how many years will the rabbit population reach its peak

This is
`t_(v)`

`t_(v) = -(b)/(2a) = -(80)/(2*(-4)) = 10`

The rabbit population will reach its peak after 10 years.