Why Cubed?Date: 11/27/2001 at 00:30:59 From: Bob Henry Subject: Why do we use cubed Why do we use cubed in the volume of a figure? Example: Find the volume of a rectangular prism of length 4ft, height 10ft, and width 5ft. I know that you would just mutiply all the numbers to get the answer 200ft. But why do they put 200ft cubed? Also why do they use a 3 to symbolize cubed? Date: 11/27/2001 at 08:44:25 From: Doctor Peterson Subject: Re: Why do we use cubed Hi, Bob. 3 What is written as ft or ft^3 in our e-mail, and read as "cubic feet," is the unit of volume equal to a 1x1x1 foot cube. You could fit 200 of these "cubic feet" into your rectangular prism. The volume of a cube is the cube (third power) of the length of a side. A 2-foot cube has volume 2^3 ft^3, or 8 cubic feet. When you multiply the dimensions, you are not just multiplying 4 x 10 x 5 ft but rather 4 ft x 10 ft x 5 ft = 4 x 10 x 5 ft x ft x ft If it makes it clearer, you can think of "4 ft" as meaning 4 times one foot. So this is actually 4 x ft x 10 x ft x 5 x ft = 4 x 10 x 5 x ft x ft x ft Just as the product of three 2's would be written as 2^3, or two cubed, the product of three "feet" is written as the cube of a foot, or ft^3. So the volume is 200 ft^3, not 200 ft. As this page points out, the older, more traditional abbreviation for a cubic foot would be cu. ft.; the form you are asking about is used by scientists, usually using the metric system. Writing 'Square Foot' http://mathforum.org/dr.math/problems/gloria.5.5.01.html Does that help? - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 11/27/2001 at 23:49:48 From: George Henry Subject: Why the word cubed Why do we use the word cubed in the answer to the volume of a cube or a rectangular prism? Why do we use the number 3 to abbreviate cubed? Date: 11/28/2001 at 08:39:21 From: Doctor Rick Subject: Re: why the word cubed Hi, George. The reason we use the word "cubed" in the answer to the volume of a cube is exactly because it is the answer to the volume of a cube. In the same way we use the word "squared" in the answer to the area of a square. If a square has sides of length 5 inches, then its area is the product of its length and width, which are both 5 inches. Therefore its area is (5 inches)*(5 inches) = 25 inches*inches (I'm using "*" for the multiplication sign.) Likewise the volume of a cube with length, width, and height equal to 5 inches is the product of these three numbers: (5 inches)*(5 inches)*(5 inches) = 125 inches*inches*inches We don't want to write "inches*inches" for the units of an area, and even more, we don't want to write "inches*inches*inches" for the units of a volume. We'd like a shorter way to write these units. And there is a way: what we call "exponents." An exponent is the number of times that you multiply by the same number. Suppose you start with 1, and multiply it by 5 four times: 1 5 = 5 = 5 (5 to the first power) 2 5*5 = 25 = 5 (5 to the second power) 3 5*5*5 = 125 = 5 (5 to the third power) 4 5*5*5*5 = 625 = 5 (5 to the fourth power) The exponent is the number above the line after the 5. When typing in e-mail, we type "5^4" for 5 to the 4th power, so we don't need to take up extra lines. When you write it, or see it in books, the exponent is usually not just above the line, but a little smaller than the base (the 5 in my examples). It's easier to write 5^4 than 5*5*5*5. There is more that we can do with exponents, but this is enough for now. It shows us an easier way to write the units of areas or volumes: inches*inches = inch^2 inches*inches*inches = inch^3 That explains why we use the little 2 for areas and the little 3 for volumes. The last thing to explain is the easiest part: why we use the words "squared" and "cubed" for these units. We could say "inches to the second power" or "inches to the third power," but again we want something shorter and easier to say. Because inches to the second power are the units of the area of a square, we call them inches squared (or square inches). Because inches to the third power are the units of the volume of a cube, we call them inches cubed (or cubic inches). Does this answer your question? I'd be glad to help you with anything else you still wonder about; we at Ask Dr. Math like to talk with kids who wonder! - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ |
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