The greatest of hair on the man's head is 104167 hairs and it occurs at 50 years.
How to determine greatest number of hair on a man's head.
This involves finding the critical points by taking the derivative of n with respect to t and setting it equal to zero.
Let's find the derivative:
dn/dt = 250t - 5t²
Set
dn/dt = 0 and solve for t
250t - 5t² = 0
250t -5t²
t(250 -5t) = 0
t = 0 or t = 250/5
t = 0 or 50
At t = 0
n = 125(0)² - 5(0)³/3 = 0
At t = 50
n = 125(50)² - 5(50)³/3
= 312500 - 208333.3
= 104167 hairs
However, since t must be in the range 0 <= t <= 75
Now, evaluate n at the critical point t = 0 and at the endpoints of the given range
At t = 0, n = 0
At t = 75
n = 125(75)² - 5(75)³/3
= 703125 - 703125
= 0.
The greatest of hair on the man's head is 104167 hairs and it occurs at 50 years.