Optimization Problems
Name:
Use what you know about determining absolute maxima and absolute minima of a
function to solve the following problems.
1.
An ordinary soft drink can has a volume of 355 cubic centimeters. Assuming this can has a
cylindrical shape, determine the dimensions (radius and height) that minimize the surface area of
the can.
2.
Determine the coordinates of point P on the graph of y  1  x 2 and having positive x-coordinate
that is closest to the origin.
3a.
A poster has a total area of 288 square inches. The poster is to have 2-inch margins on the top and
bottom and 1-inch margins on each side. What should be the dimensions of the poster so as to
maximize the area to be used for printed matter (the darkly shaded region in the figure below)?
2 in.
1 in.
1 in.
2 in.
3b.
A poster is designed to contain 200 square inches of printed matter (the darkly shaded region in the
figure above). The poster is to have 2-inch margins on the top and bottom and 1-inch margins on
each side. What should be the dimensions of the poster so as to use the least total amount of
paper?
4.
A child’s sandbox is to be made by cutting equal squares from the corners of a square sheet of
galvanized iron and turning up the sides. If each side of the sheet of galvanized iron is 2 meters
long, what size squares should be cut out of the corners to maximize the volume of the sandbox?
[Hint: draw a picture.]
5.
A ladder 8 meters long leans against a wall. At what angle should the ladder be inclined so as to
give the maximum amount of head room under the ladder at a point 2 meters from the wall?
Optimization Problems
6.
Name:
Natalie owns some land that she will sell when she needs the money. In the meantime she has
decided to rent the land to gardeners. She has found that when the rent is $250 per season, she can
rent all 200 lots, but for every $10 increase in rent, 5 plots go unrented. How much did she charge
per plot in order to maximize her income?
7. Determine the constant a so that the function f ( x)  x 2 
a
will have
x
(a) a relative minimum at x = 2
(b) a relative minimum at x = -3
(c) a relative minimum at x = 4
(d) an inflection point at x = 1
8. Find two real numbers whose difference is 42 and whose product is a minimum.
9. A fence 8 feet high is on level ground and is parallel to a building. The space between the fence and
the building is 2 feet. Find the length of the shortest ladder that will reach from the ground to the
building over the fence.
10. A lifeguard stationed on the beach (with a nice, straight shoreline) sees a swimmer in trouble 150
meters down the beach and 60 meters out in the water. The lifeguard can run 5 m/sec on the beach and
swim 3 m/sec in the water. Once the lifeguard starts running, what is the least amount of time it will
take him/her to reach the swimmer? What path along the shoreline and out into the water should the
lifeguard take to reach the swimmer in that amount of time (or, how far must the lifeguard run along the
shore before jumping into the water in order to reach the swimmer in the least amount of time)?
swimmer
60 m
jump in point
shoreline
150 m
lifeguard
Optimization Problems
Name:
11. A light is hung over the center of a square table 4 m2. The intensity of the light hitting a point P on
the table is directly proportional to the sine of the angle the path of the light makes with the table to the
table and is inversely proportional to the distance between P and the light source. How high above the
table should the light be placed in order to maximize the light intensity at the corners of table
12. Find the area of the rectangle of maximum area that can be inscribed in a
6-8-10 triangle with one side along the hypotenuse.
10
8
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