Length, area and volume: multiplying and dividing
The volume of a cylinder is calculated by the equation \(\text \pi {r}^{2}{h}\). This equation involves multiplying three lengths together
Multiplying or dividing quantities with units often occurs in everyday life. To calculate the area of a room (for the purposes of fitting a carpet, for example) you would need to multiply the length of the room by the width of a room.
Let鈥檚 look a little more closely at this:
\({length}~(m)~\times~{width}~(m)~=~{area}~{(m^2)}\)
The unit \({m^2}\) is different to \({m}\). So what we have found is that multiplying two units results in an answer in a different unit. Also, as length is one dimensional, when you multiply two one dimensional quantities together, you get a two dimensional quantity.
You should know the following in preparation for your exam:
- Number 脳 Length = Length (one dimension)
- Number 脳 Area = Area (two dimensions)
- Number 脳 Volume = Volume (three dimensions)
- Length 脳 Length = Area (two dimensions)
- Length 脳 Length 脳 Length = Volume (three dimensions)
- Length 脳 Area = Volume (three dimensions)
The opposite is also true:
Area 梅 Length = Length
Volume 梅 Number = Volume
Question
The letters \({a}\) and \({b}\) both represent lengths.
Does the following calculation make sense, and if it does, how many dimensions does the answer have?
\({a}~\times~{b^2}\)
Yes, it does make sense. \({a}\) is a length. \({b^2}\) is an area as it is a \({length}\times{length}\). A length multiplied by an area is a volume so the answer has three dimensions.
Question
The letters \({a}\), \({b}\) and \({c}\) all represent lengths.
Does the following calculation make sense, and if it does, how many dimensions does the answer have?
4\({c}~+~{a^2}\)
No, it does not make sense. 4\({c}\) is a length, and \({a^2}\) is an area. We know \({a^2}\) is an area because \({a}\) is a length and therefore \({a^2}\) is a length multiplied by a length, which is an area. You cannot add a length and an area.
Question
The letters \({a}\), \({b}\) and \({c}\) all represent lengths.
Does the following calculation make sense, and if it does, how many dimensions does the answer have?
\(\frac {a^2} {b}~+\)4\({c}\)
Yes, it does make sense. \(\frac {a^2} {b}\) is an area divided by a length, which is a length. 4\({c}\) is a number multiplied by a length, which is a length.
Length + Length = Length, so the answer has one dimension.
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