# Constructing a Closed Box

• Nov 4th 2008, 06:35 AM
magentarita
Constructing a Closed Box
A closed box with a square base is required to have a volume of 10 cubic feet.

(a) Express the amount A of material used to make such a box as a function of the length x of a side of the square base.

(b) How much material is required for a base 1 foot by 1 foot?

• Nov 5th 2008, 10:37 AM
Soroban
Hello, Magentarita!

Quote:

A closed box with a square base is required to have a volume of 10 ft³

(a) Express the amount $A$ of material used to make such a box
as a function of the length $x$ of a side of the square base.

Code:

        *-----*         /    /|       /    / | y       *-----*  |       |    |  *     y |    | /       |    |/ x       *-----*         x

The volume is 10 ft³: . $V \:=\:x^2y \:=\:10 \quad\Rightarrow\quad y \:=\:\frac{10}{x^2}$ .[1]

The surface area is: . $A \;=\;2x^2 + 4xy$ .[2]

Substitute [1] into [2]: . $A \;=\;2x^2 + 4x\left(\frac{10}{x^2}\right)$

Therefore: . $\boxed{A \;=\;2x^2 + \frac{40}{x}}$

Quote:

(b) How much material is required for a base 1 ft by 1 ft?
If $x = 1\!:\;\;A \;=\;2(1^2) + \frac{40}{1} \;=\;\boxed{42\text{ ft}^2}$

• Nov 5th 2008, 12:40 PM
magentarita
Wonderfully done
Quote:

Originally Posted by Soroban
Hello, Magentarita!

Code:

        *-----*         /    /|       /    / | y       *-----*  |       |    |  *     y |    | /       |    |/ x       *-----*         x
The volume is 10 ft³: . $V \:=\:x^2y \:=\:10 \quad\Rightarrow\quad y \:=\:\frac{10}{x^2}$ .[1]

The surface area is: . $A \;=\;2x^2 + 4xy$ .[2]

Substitute [1] into [2]: . $A \;=\;2x^2 + 4x\left(\frac{10}{x^2}\right)$

Therefore: . $\boxed{A \;=\;2x^2 + \frac{40}{x}}$

If $x = 1\!:\;\;A \;=\;2(1^2) + \frac{40}{1} \;=\;\boxed{42\text{ ft}^2}$

I love your answers to my questions. I will be taking calculus 1 next semester. I hope you are there for me.