Calculus problems >______<?

Question

1) A club sells memberships for $420 if 50 people or fewer join. For each person in excess of 50 who joins, the memebership fee of each member is reduced by $2. The club can accept no more than 150 members. How many memberships should be sold to maximize the club's recipts?

2) A rectangular sheet of paper is to be used for a rectangular printed notice. The page must contain 30 square inches of print and there must be margins of 2 inches on each side and 1 inch on the top and bottom. Find the dimensions of the sheet of smallest area which will suffice.

Answer 1

1)
The income of the club is
n*(420-2*(n-50))
where n is the number of members.
It is maximum for n=130.

2)
Let x is the height of the printed area.
The width of the printed area is 30/x.
The height of the sheet is (x+2).
The width of the sheet is (30/x + 4).
The size of the sheet is
(x+2)* (30/x + 4) = 4*x + 60/x + 38
Its minimum is for x = 3.873 inches.
30/x = 7.746 inches.
The height of the sheet is 53.873 inches.
The width of the sheet is 11.746 inches.
-
Extremes calculated in both cases as critical points of the function. (First derivative equals to zero.)
-