Application of quadratic equation

Question

The population P(t) a culture of bacteria is given by P(t) = - 1731t ^ 2 + 88, 000t + 10, 000 where 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

Recognize the equation as a quadratic, whose graph is a parabola. The fact that the first term is negative means that it opens downward. The maximum occurs at the vertex, which is always on the vertical line

x = -b/2a.

In this case that would be x = (-88000)/(-1731) ≈ 51 hours