The population P(t) of a culture of bacteria is given by P(t)=-1873t^2+95,000+15000, 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.

We have been given that the population P(t) of a culture of bacteria is given by [tex]P(t)=-1873t^2+95,000t+15000[/tex], where t is the time in hours since the culture was started. We are asked to find the time at which the population is at a maximum.

We can see that our given function is a downward opening parabola, so it maximum point will be vertex. To find the time at which the population is at a maximum, we will find the x-coordinate of parabola.

We know that x-coordinate of a parabola can be found using formula [tex]x=\frac{-b}{2a}[/tex].

[tex]x=\frac{-95,000}{2(-1873)}[/tex]

[tex]x=\frac{-95,000}{-3746}[/tex]

[tex]x=25.36038[/tex]

Upon rounding our answer to nearest hour, we will get:

[tex]x\approx 25[/tex]

Therefore, the population will be maximum at 25 hours.