posted by .

A closed box is to be rectangular solid with a square base. If the volume is 32in^3, determine the dimensions for which the surface area is minimum.

  • Calculus -

    make a sketch
    let the base be x by x, and the height be y
    so x^2 y = 32
    y = 32/x^2

    SA = 2x^2 + 4xy
    = 2x^2 + 4x(32/x^2)
    = 2x^2 + 128/x

    d(SA)/dx = 4x - 128/x^2 = 0 for a min of SA
    4x = 128/x^2
    4x^3 = 128
    x^3 = 32
    x = 32^(1/3) or appr 3.175
    y = 32/3.175^2

    Well , what do you know, it happens to be a perfect cube.

  • Calculus -


Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. Calculus

    I actually have two questions: 4. An open box is to be made from a rectangular piece of material 3m by 2m by cutting a congruent square from each corner and folding up the sides. What are the dimensions of the box of the largest volume …
  2. calculus

    determine the dimensions of a rectangular solid (with a square base) with maximum volume if it's surface area is 289 meters
  3. calculus

    A container company is tasked to make an open-top rectangular box with a square base. The box must have a volume of 108cm^(3). let the length of the sides of the square base be x cm and the height h cm. (1) what value of x will produce …
  4. Calculus

    Q. An open-top box with a square base is to have a volume of 62.5 square inches. What dimensions will yield the box of minimum surface area?
  5. Calculus

    A closed box is to be a rectangular solid with a square base and volume of 12 ft^3. Find the most economical dimensions if the top of the box is twice as expensive as the sides and bottom.
  6. Calculus

    A cardboard box of 32in^3 volume with a square base and open top is to be constructed. What is the length of base that will minimize the surface area?
  7. Calculus

    A rectangular box is to have a square base and a volume of 20 ft3. If the material for the base costs 37¢/square foot, the material for the sides costs 10¢/square foot, and the material for the top costs 13¢/square foot, determine …
  8. calculus

    a. A closed cylindrical can is to hold 1000cm^3 of liquid. How should we choose the height and radius of this can to minimize the amount of material need to manufacture the can?
  9. Calculus

    A box with a square base and no top is to have a volume of 32 ft3. What dimensions use the least amount of material (in other words what dimensions give minimum outside surface area)?
  10. Math

    We want to construct a closed rectangular box whose base has a length three times the width. The surface area has to be 600 cm2. Determine the dimensions of the box that will produce the largest volume.

More Similar Questions