Area and Volume of CuboidDate: 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%? Please help me. 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/ |
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