# Thread: Surface area as function of height of box

1. ## Surface area as function of height of box

A box with a square base (sides of length x) and an open top must have a volume of 32,000 cubic centimeters. Express the surface area as a function of the height of the box

This is all I got...

l x w x h = 32,000

x = w, h

h = 32,000/x^2 since Volume = x^2(h)

2. Hello, realintegerz!

Did you make a sketch?

A box with a square base (sides of length $x$) and an open top
must have a volume of 32,000 cubic centimeters.
Express the surface area as a function of the height of the box.
Code:
         *-------*
/|      /|
/ |     / |
*-------*  |
|       |  |
|       |  |
h|       |  *
|       | /
|       |/ x
*-------*
x

The volume is: . $x^2h \:=\:32,\!000 \quad\Rightarrow\quad x^2 \:=\:\frac{32,\!000}{h} \quad\Rightarrow\quad x \:=\:\frac{80\sqrt{5}}{\sqrt{h}}$ .[1]

The surface area is comprised of:

. . a square base with area: $x^2$

. . four side panels, each with area: $xh$

The total surface area is: . $A \;=\;x^2 + 4xh$

Substitute [1]: . $A \;=\;\frac{32,\!000}{h} + 4\left(\frac{80\sqrt{5}}{\sqrt{h}}\right)h$

Therefore: . $\boxed{A \;=\;\frac{32,\!000}{h} + 320\sqrt{5h}}$