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### Area and Volume of Cuboid

```Date: 09/09/2002 at 23:28:58
From: Safwan
Subject: Area and Volume of cuboid

Hello Dr. Math,

I have questions about the area and the volume of a cuboid.

The length of the cuboid is 12cm, its width is 7cm, and its height is
3cm.  How can I find the new volume and the surface area of this
cuboid after its volume has been reduced 10%?

```

```
Date: 09/10/2002 at 17:28:24
From: Doctor Ian
Subject: Re: Area and Volume of cuboid

Hi,

Let's think about an arbitrary cuboid. We'll use L, W, and T to stand
for the length, width, and thickness.

Also, because it's easier to work with, let's reduce the volume to 1/8
of its original size. The volume of the cuboid is

volume = LWT

and the surface area is

area = 2(LW + LT + WT)

Now, we can make the volume 1/8 of its original value by making any of
the dimensions 1/8 of its original value:

volume' = (L/8)WT

= (1/8)(LWT)

= (1/8)volume

So that works.  What does this do to the area?

area' = 2((L/8)W + (L/8)T + WT)

= 2(LW + LT - WT - (7/8)LW - (7/8)LT)

= 2(LW + LT + WT) - (7/4)LW - (7/4)LT

= area - (7/4)(LW + LT)

Now, the question is, is this the _only_ way to reduce the volume to
1/8 of its original value? No. Another way would be to reduce each
dimension by 1/2:

volume' = (L/2)(W/2)(T/2)

= (1/2)(1/2)(1/2)LWT

= (1/8)LWT

= (1/8)volume

Now, what does this do to the area?

area' = 2((L/2)(W/2) + (L/2)(T/2) + (W/2)(T/2))

= 2((1/4)LW + (1/4)LT + (1/4)WT)

= (1/4) * 2(LW + LT + WT)

= (1/4)area

Which is different from what we got before. So in order to know what
happens to the area, it's not enough to know that the volume has been
reduced by some amount. You have to know _how_ it was reduced. Does
this make sense?

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Higher-Dimensional Geometry

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