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

Answer:

The population is at a maximum after 22 hours.

Step-by-step 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

$x_{v} = -\frac{b}{2a}$

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

$t_{v} = -\frac{81000}{2*(-1840)} = 22$

The population is at a maximum after 22 hours.