Calculus 1 - Optimization Problems
1.
An open box with a rectangular base is to be constructed from a rectangular piece of cardboard 16 in.
wide and 21 in. long by cutting out a square from each corner, and then bending up the sides. Find the
size of the corner square which will produce a box having the largest possible volume.
2.
An open box is to be made from a piece of cardboard 12 in. square by cutting out a square from each
corner and bending up the sides. Find the size of the corner square which will produce a box having
the largest possible volume.
3.
If a box with a square base and open top is to have a volume of 4 ft.3, find the dimensions that require
the least material.
4.
A page of a book is to have an area of 90 in.2 with 1 in. margins at the bottom and sides, and a ½ in.
margin at the top. Find the dimensions of the page that will allow the largest printed area.
5.
A circular cylindrical container, open at the top, and having a capacity of 24 ∏ in.3 is to be made. If
the cost of the material for the bottom of the container is three times that used for the curved part, find
the dimensions that will minimize cost.
6.
A rectangular field is to be fenced off along a river and no fence is required along the river. If the
material for the fence costs $4.00 per foot for the ends and $6.00 per foot for the side parallel to the
river, find the dimensions of the largest possible area that can be enclosed with $1,800.00 worth of
fence.
7.
A North–South highway intersects with an East–West highway at P. An auto crosses P at 10:00 AM,
traveling East at 20 mph. At the same instant, another car is 2 mi. North of P, traveling South at 50
mph. Find the time they are closest to each other.
8.
A window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is
15 ft., find the dimensions that will allow the maximum amount of light to enter.
9.
At 1:00 PM, Ship A is 30 mi. due South of Ship B, and is sailing North at 15 mph. If B is sailing
West at 10 mph, find the time at which the distance between the ships is minimal.
10.
A veterinarian has 100 ft. of fencing and wishes to construct five dog runs by first building a fence
around a rectangular region, and then subdividing that region into five smaller rectangles by placing
four more fences in parallel. What dimensions of the region will maximize the total area?
11.
A rectangular plot of land will be bounded on one side by a river and on the other three sides by a
single–strand electric fence. With 800 ft. of wire, what is the largest area you can enclose?
12.
A package can be sent by parcel post only if the sum of its length and girth (the perimeter of the base)
is not more than 96 in. Find the dimensions of the box of maximum volume that can be sent if the
base of the box is square.