Posted by Rob on Wednesday, March 7, 2007 at 11:25am.
Since the agency only takes groups of no less than 30 people, the minimum profit for a single trip is $15*30 people.
Let x be the number of people above 30.
The number of people is then 30+x.
The [Total Revenue] is equal to the
[price charged for each person] times the [number of people]
[price charged for each]=$15-.15*x
Notice if x=0, then the price charged is equal to $15 and the number of people in group is 30, which is the minum group size the company will allow for a trip.
Thus our Total Revenue Function is
[Total Revenue] =
[price charged for each person]*[number of people]
=($15-$.15*x)*(30+x)
R(x)= ($15-$.15*x)*(30+x)=450+10.5x-.15*x^2 after foiling everything out.
R'(x)=10.5-2*.15*x is the derivative of the Revenue function and the Revenue will be maximized at a value of x for which R'(x)=0.
Setting R'(x)=0 implies 10.5-2*.15*x=0, which implies that x=35 people. The group size was constrained to be at least 30 people and at most 50 people, and since the number 35 is between 30 and 50, then a group size of 35 people is a valid solution for this problem: according to the revenue function R(x),and the constraints on groups sizes given above above, the company will maximize its profit when it has groups of size 35. Now I must go breath.