Practice setting up optimization problems
For the following problems, get to the point where you have an equation for the quantity being optimized in
terms of a single variable, and identify the domain for that variable. Solutions on the back...
(1) What point on the graph of f (x) = x2 + 2x + 1 lies closest to the point (2, 1)?
(2) What is the area of the largest rectangle that can be inscribed in the unit circle?
(3) What are the dimensions of the largest open-top box that can be made from a 3” by 5” card by cutting
squares out of each corner and folding the sides up?
(4) I have 1000 feet of fencing, and want to build a rectangular pen along a cliff wall with maximum area. The
cliff wall is perfectly straight, and acts as one side of the fence. What dimensions of pen should I make?
(5) I want to build a rectangular fence with three sides made from chain-link fence, at two dollars per foot, and
one side from wooden pickets, at ten dollars per foot. I want the area inside the fence to be 10,000 square
feet - what are the dimensions that will cost the least to build?
(6) A concert venue can hold 18,000 people. When concert tickets are priced at 15 per person, 10, 000 tickets
are sold, and when they are priced at 12 a person, 13, 000 tickets are sold. Assume a straight-line function
describes number of tickets sold at a given price. What ticket price will maximize revenue for a concert?
(7) What is the radius of the right circular cylinder of largest volume that can be inscribed in a sphere of
radius 5?
(8) Two ships are sailing in open water. The first ship’s (x, y) position is given by x(t) = 10 sin t,
y(t) = 20 cos t. The second ship’s position is given by x(t) = 3t, y(t) = 4t. At what time are the ships
nearest one another and how close are they to each other at that time?