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Students decided to construct a box for the Monopoly game such that its width is

exactly three times its length and its volume is 36 cubic centimeters.

Assuming that the box is closed on all sides and using calculus methods only,

determine the smallest amount of material needed for its construction. Show all

your work.

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MrRoyal

The smallest amount of material needed is 54 square centimeters

How to determine the amount of material needed?

The given parameters are:

Volume = 36 cubic centimeters

Represent length with x, width with y and height with z.

So, we have

x = 3y

The volume is calculated as:

V = xyz

This gives

V = 3y²z

Substitute 36 for V

3y²z = 36

Divide by 3

y²z = 12

Make z the subject

z = 12/y²

The surface area is:

S = 2(xy + xz + yz)

This gives

S = 2(3y² + 3yz + yz)

Evaluate the like terms

S = 2(3y² + 4yz)

Expand

S = 6y² + 8yz

Substitute z = 12/y²

S = 6y² + 8y * 12/y²

This gives

S = 6y² + 96/y

Differentiate

S' = 12y -  96/y²

Set to 0

12y -  96/y² = 0

Multiply through by y²

12y³ - 96 = 0

Add 96 to both sides

12y³ = 96

Divide by 12

y³ = 8

Take the cube root of both sides

y = 2

Recall that:

x = 3y and z = 12/y²

This gives

x = 3 * 2 = 6

z = 12/2² = 3

Recall that:

S = 2(xy + xz + yz)

So, we have:

S = 2(6 * 2 + 3 * 3 + 2 * 3)

Evaluate

S = 54

Hence, the smallest amount of material needed is 54 square centimeters

Read more about surface areas at:

https://brainly.com/question/76387

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