posted by .

A box with a square base and an open top is constructed from 5400 cm^2 of cardboard. Find the dimensions of the largest possible box.

I know the answer is :
base lenght - 42.4 cm
height- 21.2 cm

please help me, thank you so much :)

  • Pre-Cal -

    It depends on the shape of the 5400cm^2 cardboard.

    If it is square, it is sqrt5400 or 73.4cm to a side.

    Now set it down, call the sides s, and the height h.
    You have to cut out h^2 from each of the four corners.

    Area=b^2 h
    but 2h+b=73.4 or h=(73.4-b)/2

    dArea/db= 2b(73.4-b)/2 -b^2/2=0
    solve for b

    Then use that to go back and solve for h.

  • Pre-Cal -

    If this is the typical questions, identical squares are to be cut out of each of the corners of the cardboard, and then the sides folded up to form the box.

    Let each side of the cutout be x cm
    so each side of the base of the box will be √5400 - 2x
    volume of box
    = x(√5400-2x)(√5400-2x)
    = 5400x-4√5400x^2 + 4x^3

    d(volume)/dx = 5400 - 8√5400x + 12x^2 = 0 for a max of volume

    x^2 - 20√6 + 450 = 0

    using the quadratic formula

    x = (20√6 ± √(2400 -4(1)(450))/2
    = (20√6 ± √600)/2
    = 36.74 or 12.25

    The first answer gives us a volume of zero, the minimum, and the second gives us the maximum value

    height = 12.25
    base = √5400-2x = 48.99

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. PreCal

    A square sheet of cardboard 18 inches is made into an open box (there is no top), by cutting squares of equal size out of each corner and folding up the sides. Find the dimensions of the box with the maximun volume. Volume= base(width)height …
  2. calculus

    A rectangular box is to be constructed from two different materials. The box will have a square base and open top. The material for the bottom costs $4.25/m2. The material for the sides costs $2.50/m2. Find the dimensions of the box …
  3. Pre Cal

    find the surface area of a box of hieght h whose base dimensions are p and q, and that satisfies either one of the following conditions: a) the box is closed. b) the box has an open top. c) the box has an open top and a square base.
  4. calculus

    This is problem 16 Section 4.6 page 246. A closed box with square base is to be built to house an ant colony. The bottom of the box and all four sides are to be made of material costing dollar/sq ft, and the top is to be constructed …
  5. 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?
  6. Physics

    An open box with a square base (see figure) is to be constructed from 160 square inches of material. The height of the box is 3 inches. What are the dimensions of the box?
  7. MATH

    A box with an open top has vertical sides, a square bottom, and a volume of 4 cubic meters. If the box has the least possible surface area, find its dimensions. Length of base= Height=
  8. Calculus 1-Optimization

    A box with a square base and open top must have a volume of 4,000 cm^3. Find the dimensions of the box that minimize the amount of material used. sides of base cm height cm
  9. pre calc

    A cardboard box has a square base and a square top. The height of the box is 13 inches. Express the surface area A (the sum of the areas of all six sides of the box) in terms of the length l of the box. Can someone please help me
  10. Math - Calculus

    You are given 1200 cm^2 of cardboard to make a box with a square base and an open top. Find the largest possible volume of the box.

More Similar Questions