Exercises - Optimization

  1. A rectangular, open-topped, square-bottomed aquarium is to hold 1 cubic meter of water. Given that the material for the bottom costs twice as much per square meter as that for the sides, find the dimensions of the least expensive aquarium.

  2. A page is to contain 30 in^2 of print. The margins at the top and the bottom of the page are each 2 in wide. The margins on each side of the page are only 1 in wide. Find the dimensions of the page so that the least amount of paper is used.

  3. A cylindrical container is to be constructed to hold $32\pi$ cm3 of flammable material. The cost per cm2 of constructing the top is three times that of the sides and bottom, since the top must contain a safety valve. What dimensions will minimize cost of construction?

  4. Rectangular boxes are formed by cutting squares of equal sizes from the corners of rectangular sheets of cardboard and by folding up the remaining material. Find the dimensions of the box of largest volume which can be formed from a 5 ft $\times$ 8 ft sheet of cardboard.

  5. A box with no top has base length twice its width. Find the box of maximum volume made from 24 square feet of material.

  6. Let $x$ and $y$ be legs of a right triangle whose hypotenuse is 2. For what values of $x$ and $y$ is $2x+y^2$ a maximum?

  7. A rectangle box is to have a square base and top and is to hold 1000 in3 of material. What are the dimensions of the box if it is to have a minimum surface area?

  8. A window in the shape of a rectangle capped by a semicircle is to be surrounded by $p$ inches of metal border. Find the radius of a semicircular part if the total area of the window is to be a maximum.

  9. A rectangular garden is to be laid out along a neighbor's property line and is to contain 432 yds2. If the neighbor pays for half of the dividing fence, what are the dimensions so that the cost to the owner is minimized?

  10. A rectangular box with top has base length three times its width. Find the dimensions of such a box of maximum volume that can be constructed from 200 in2 of material.

  11. A rectangle has an area of 8 in2. Find the dimensions so that the distance from one corner to the midpoint of the opposite side is minimized.

  12. A museum display case in the shape of a rectangular box with square base and a volume of 54 ft3 is to be built. The front, back, and sides are to be made of glass, which costs $\$$10 per square foot; the top and bottom are to be made of fine wood, which costs $\$$20 per square foot.

    1. Find the dimensions of the least expensive case that can be so constructed.
    2. Find the cost of the least expensive case.