# relateing surface area to volume?

• Sep 5th 2009, 11:53 AM
diver15
relateing surface area to volume?
I have a problem that I can't quite figure out.

If I have a rectangular prism with a length=L width=W and hight=H and L=W, and the volume of this prism is 10,000 then you could write the volume formula as so

L(W)H=10,000

X^2(H)=10,000 because L=W and X=L=W X^2=L(W)

and surface area of the prism could be writen as so

2LW+2LH+2HW= surface area

2X^2 +2XH+2XH= surface area, Because X^2=L(W) and X=L=H

2X^2+4X= surface area

Now I have to find out what the minimum surface area would be. Do I have to use some linear optimization proces on a calculater, If so can somone give me a quick pointer, I have a TI-83. And if not, can somone show me the error of my way?
• Sep 5th 2009, 12:53 PM
galactus
Since L=W, we can write the volume as $L^{2}H=10,000$..[1]

Assuming there is a top, the surface area is $S=2L^{2}+4LH$...[2]

If we want to minimize the surface area, we can solve the volume formula for H and sub into S.

From [1], $H=\frac{10,000}{L^{2}}$...[3]

Sub into [2]:

$S=2L^{2}+4L(\frac{10,000}{L^{2}})$

$S=2L^{2}+\frac{40,000}{L}$

Differentiate, set to 0 and solve for L.

$\frac{dS}{dL}=4L-\frac{40,000}{L^{2}}$

$4L-\frac{40,000}{L^{2}}=0$

$L=10\cdot 10^{\frac{1}{3}}\approx 21.544$

The width is the same. Plug this into [3] to find the height.

We get $H=21.544$.

It's the same. The minimum surface area occurs when we have a cube.

That means the surface area is $S=6L^{2}=2784.953$

Ti-83's do not do derivatives. For that get a TI-89, Voyage 200, etc.
• Sep 5th 2009, 02:42 PM
diver15
what if only the base were square and the prism could not be a cube
• Sep 5th 2009, 03:21 PM
galactus
The point is if the base is square the minimum surface area is achieved when we have a cube. That is, when the height, width, and height are the same.If the base were not square, then that's another matter.

Another example is when we want to find the minimum surface area of a cylinder. That is achieved when the height and diameter are equal.