CONCEPT: Functions and derivatives
At the point maximum, f'(x)=0

SOLUTION: Given, the number of people per trip is x.

The fare for each person =[(54−(x/32)]^2

The revenue from x people =x[(54−(x/32)]^2 =g(x)

where g(x)=[x(54−(x/32)]^2

When the revenue is the maximum, the marginal revenue equal to zero.
=>g′(x)=[54−(x/32)]^2−(2x/32)[(54-(x/32)]

For the maximum value of g(x):
g′(x)=0
 54−(x/32)=0 and (54−3x/32)=0
 x=1728,x=576

 Given the capacity of the train is 900, the number of people per trip = 576 (C)