Finding the Largest Possible Volume

Question

Suppose you want to build a box,and you have 100 square feet of cardboard material to use.You want the box to have as much volume as possible.Find the whole number dimensions that will produce the greatest volume.What is this volume?What is this volume? Show all calculations.

Answer 1

if the box has 2 sides, 2 bottoms and 2 tops (i.e. it does not have an open top), the the surface area of the box = 2xy + 2xz + 2yz = 100 ft²

The volume of the box is xyz. To minimize the volume with a constraint, Use Lagrange Multipliers

Let the function be V = xyz + W(2xy + 2xz + 2yz - 100) where W is the Lagrange variable

Taking the derivatives:

V_x = yz + W(2y + 2z), V_y = xz + W(2x + 2z), V_z = xy + W(2x + 2y), V_w = (2xy+2xz+2yz-100)

Setting each equal to zero you get and manipulating the variable for the first 3 equations, you get

-yz = W(2y+2z) -xz = W(2x+2z) -xy = W(2x+2y)

Now multiply the first equation by x, the second by y & the third by z to get

-xyz = W(2xy+2xz) -xyz = W(2xy+2yz) -xyz = W(2xz+2yz)

Settting the W terms equal to each other (you can do the algebra), you will find that

x = y = z which is what you would expect!

So, for the surface area, just use the x variable to solve for the numeric value like so 2x² + 2x² + 2x² = 100 or 6x² = 100 or x = 4.0825 ft

That means that the volume is (4.0825)³ = 68.04 ft3

Checking the surface area, 2(4.085)² + 2(4.085)² + 2(4.085)² = 100 ft² !

Answer 2

The box with the largest possible volume will always be a cube, so you are looking for the cube that has 100 square feet of surface area. 100/6 is 16.67 square feet. So you want a square with that surface area, which means you want the square root of 16.67, which is 4.082 feet. You want a cube with sides of 4.082 feet.