Question

The population P(t) of a culture of bacteria is given by P(t) = -1,840t2 + 81,000t + 10,000, where t is the time in hours since the culture was started. Determine the time at which the population is at a maximum. Round to the nearest hour.

Answer 1

The population is at a maximum after 22 hours.

Explanation:

Vertex of a quadratic function:

Suppose we have a quadratic function in the following format:

`f(x) = ax^(2) + bx + c`

It's vertex is the point
`(x_(v), f(x_(v))`

In which

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

If a<0, the vertex is a maximum point, that is, the maximum value happens at
`x_(v)`, and it's value is
`f(x_(v))`

In this question:

`P(t) = -1840t^(2) + 81000t + 10000`

Determine the time at which the population is at a maximum.

This is the value of t at the vertex.

We have that
`a = -1840, b = 81000`. So

`t_(v) = -(81000)/(2*(-1840)) = 22`

The population is at a maximum after 22 hours.