# Math Help - Constructing a Closed Box

1. ## 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?

2. Hello, Magentarita!

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

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

3. ## Wonderfully done

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.