The revenue per person is $200 for 50 minutes. If the number of persons is greater than 50 but less than 80 the revenue earned per person is reduced by $2 to $198. For x number of people the cost incurred is $6000 + 32*x
If the number of people x is `x <= 50` , the profit earned is $200*x - 6000 - 32*x = -6000 + 168*x and for `80 >= x >50` , the profit earned is 198*x - 6000 - 32*x = -6000 + 166*x
From the expressions for profit derived earlier it is seen that as the number of people increases there is an increase in profit as -6000 is a constant and both of 168*x and 166*x are always positive.
For x = 50, the profit earned is $2400 and for x = 80, the profit earned is $7280.
The number of people to maximize profit is 80.