Answer: Let's declare some variables to make this problem easier to solve:
x: the number of extra people above the initial 60 attendees
p: the price per person after the discount is applied
Using this notation, we can express the price per person as:
p(x) = 30 - (x/10) * 1.5
Note that the price per person decreases by $1.50 for every 10 extra people, which is equivalent to a decrease of $0.15 per person.
The total number of attendees will be 60 + x, and the total revenue generated will be:
R(x) = p(x) * (60 + x)
We want to find the price per person that results in the greatest revenue. To do this, we need to find the maximum value of the revenue function R(x). We can do this by taking the derivative of R(x) with respect to x, setting it equal to zero, and solving for x:
R'(x) = (30 - (x/10) * 1.5) * 1 + (60 + x) * (-1/10 * 1.5)
R'(x) = 30 - 0.15x - 9 - 0.15x
R'(x) = -0.3x + 21
-0.3x + 21 = 0
x = 70
Therefore, to maximize revenue, the hall should have 130 attendees (60 initial attendees + 70 extra attendees). The price per person in this case would be:
p(70) = 30 - (70/10) * 1.5 = $21.00
So the hall should charge $21.00 per person to result in the greatest revenue.
Explanation: