A travel agency offers group tours at a rate of $60/person for the first 30 participants. For longer groups-up to 90-each person receives a$0.50 discount for every participant in excess of 30. For example, if 31 participate, the cost/person is $59.50. Determine the size of the group that will produce the maximum amount of money for the agency.

revenue for x>=30

r = x(60-0.50(x-30))
= .5x(150-x)

max r at x=75

To determine the size of the group that will produce the maximum amount of money for the agency, we need to find the point where the discount savings equal the increase in participants.

Let's assume the size of the group is x.

For the first 30 participants, the rate is $60/person. So, the cost for the first 30 participants is 30 * $60 = $1800.

For any additional participant beyond 30, a $0.50 discount is given for each participant. So, for x participants (x > 30), the cost per person would be $60 - ($0.50 * (x - 30)).

The total cost for x participants would be (x * ($60 - ($0.50 * (x - 30)))).

To find the maximum amount of money for the agency, we need to determine the value of x that maximizes the total cost.

We can set up this problem as a function, where the total cost (C) is a function of the number of participants (x):

C(x) = x * ($60 - ($0.50 * (x - 30)))

Now, let's find the maximum point of this function by taking its derivative and setting it equal to zero:

C'(x) = 0

To find the derivative, we can apply the product rule:

C'(x) = (($60 - ($0.50 * (x - 30))) * 1) + (x * (-0.50))

Simplifying the equation:
C'(x) = ($60 - ($0.50 * (x - 30))) - 0.50x

Setting C'(x) equal to zero:
($60 - ($0.50 * (x - 30))) - 0.50x = 0

Now, we can solve for x:

($60 - ($0.50 * (x - 30))) = 0.50x

$60 - ($0.50x - 15) = 0.50x

$60 - $0.50x + $15 = 0.50x

$75 = $1x

x = 75

Therefore, the size of the group that will produce the maximum amount of money for the agency is 75 participants.