PLEASE SHOW ALL WORK my teacher only cares about the work no

PLEASE SHOW ALL WORK..... my teacher only cares about the work, not the answer!

A secret society charges its members $200 per year. For each membership increase over 60 members, the membership cost is decreased by $2.00. What number of members lead to the maximum revenue for the club?

Solution

Let x be equal to the number of members above 60.

Set up an equation that multiples the number of members to the cost of membership.

The total number of members then will be (X + 60)

The cost of membership will then be (200 - 2*X) (since it decreases by $2 for each member over 60).

Multiply the two together and you get

total cost

200*X - 2*X² + 12000 - 120*X

which can be reduced to

2*X² - 80*X - 6000

Take the first derivative of this and set that equal to zero (to maximize the cost)

4*X - 80 = 0

Solve for X and you get

X = 20

Thus, you maximize your profits when you have 20 members more than 60

So the total number of members that maximizes revenue is 60 + 20 = 80 members

Test it out.

80 members means each membership costs $160 which gives $12,800.
79 members means each membership costs $162 which gives $12,798
81 members means each membership costs $158 which gives $12,798