the population P (t) of a culture of bacteria is given by P (t) = -1710t + 92,000t+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.

The question might have some mistake since there are 2 multiplier of t. I found a similar question as follows:

The population P (t) of a culture of bacteria is given by P (t) = - 1710t^2 + 92,000t + 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:

27 hours

Step-by-step explanation:

Equation of population P (t) = - 1710t^2 + 92,000t + 10,000

Find the derivative of the function to find the critical value

dP/dt = - 2 (1710) t + 92000

= - 3420t + 92000

Find the critical value by equating dP/dt = 0

-3420t + 92000 = 0

92000 = 3420t

t = 92000/3420 = 26.90

Check if it really have max value through 2nd derivative

d (dP) / dt^2 = - 3420

2nd derivative is negative, hence it has maximum value

So, the time when it is maximum is 26.9 or 27 hours