Math problem....help plz??

Question

At a concert the audience enter the venue at a rate given by:
dN/dt = 10t - t^2/12 until all audience has been admitted. N is number of people admitted and t is minutes after opening.

How long does it take to admit all the audience?
How many people were admitted?

Answer 1

Assuming "admit all the audience" means "when the rate of admission reaches zero", that's:

dN/dt = 0
10t - t^2/12 = 0
t(120 - t) = 0

t=0 (can be ignored) or t=120 minutes. 2 hours is the answer.

The number of people admitted at that point is the integral from 0 to 120 of dN/dt:

N(t) = 5t^2 - t^3/36 + C

N(120) - N(0) = 5*120^2 - 120^3/36 + C - 0 - 0 - C
N(120) - N(0) = 72,000 - 48,000
N(120) - N(0) = 24,000

Answer 2

N=Integral(10t-t^2/12)
N=5t^2-t^3/36+C
Since N=0 when t=0, C=0
So N=5t^2-t^3/36
All people have been admitted when dN/dt=0 again (after t=0)
10t-t^2/12=0
t=120
N=5(120^2)-120^3/36=24,000