Math 1261: Calculus I
Brown
Optimization Problems
1. Find a positive number such that the sum of it and its reciprocal is as small as possible.
2. A box with a square base and open top must have a volume of 32,000 cm3 . Find the dimensions of the
box that minimize the amount of material used.
3. A piece of wire 3 cm in length is cut into two pieces. One piece is bent into the shape of a circle and
the other is bent into the shape of a square. Determine how to obtain the maximum area enclosed by
the two figures.
4. Find the area of the largest rectangle that can be inscribed in a right triangle with legs of length 3 cm
and 4 cm if two sides of the rectangle lie along the legs.
5. A right circular cylinder is inscribed in a cone with height h and base radius r. Find the largest possible
1
volume of such a cylinder. (Vcone = πr2 h and Vcylinder = πr2 h)
3
6. A woman at a point A on the shore of a circular lake with radius 2 miles wants to arrive at a point C
diametrically opposite A on the other side of the lake in the shortest possible time. She can walk at a
rate of 4 miles per hour and row a boat at a rate of 2 miles per hour. How should she proceed?
7. For the given cost and demand functions, find the production level that will maximize profit.
C(x) = 680 + 4x + 0.01x2 ,
p(x) = 12 −
x
500
8. Find the production level at which the marginal cost function starts to increase.
C(x) = 0.0002x3 − 0.25x2 + 4x + 1500
9. A baseball team plays in a stadium that holds 55,000 spectators. With ticket prices at $ 55, the average
attendance had been 27,000. When ticket prices were lowered to $ 8, the average attendance rose to
33,000. Find the demand function, assuming it is linear, and determine how much the team should
charge to maximize revenue.
Note: Some useful relationships for economic and business application. P , C, R, and p denote profit,
cost, revenue, and demand functions.
P (x)
= R(x) − C(x),
Profit=Revenue-Cost
R(x)
= xp(x),
Revenue=(units sold)(price per unit)
c(x)
=
C(x)
,
x
Average Cost = (Cost per unit)/(units sold)