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Dimensions in Formulae C = 2 r r r r A = r 2 These formulae make sense since they are measures of one, two and three dimensional quantities respectively. So if: r length r x r area (length x length) r x r x r volume (length x length x length) This is true in general for any formulae so if: a, b and c are lengths then: a length a x b area a x b x c volume
Dimensions in Formulae We can use dimensions to check any formula that we are using (or have derived) as being reasonable. Consider the formulae for the circumference and area of a circle, as well as the volume of a sphere shown below. From the formulae you can see that: C = 2 rA = r 2 rrr r r2r2 r3r3 (1) the length of the circumference of a circle is a multiple of its one dimensional radius. C = 2 r A = r 2 r2r2 (2) the area of a circle is a multiple of the 2 dimensional square on its radius. (3) the volume of a sphere is a multiple of the 3 dimensional cube on its radius. r3r3
Can you work out this formula by yourself? A sphere of radius r, just sits inside a hollow cylinder. Nicola tried to derive a formula for the volume of the space not occupied by the sphere. She ended up with 3 different formulas only one of which was correct. Which formula is correct and why? C = 2 r r r r A = r 2
Dimensions in Formulae C = 2 r r r r A = r 2 Any multiple of these will simply increase or decrease the size of the dimension involved, depending on the value of the multiplier. If a, b and c are lengths and k is a multiplier then: ka length k(a x b) area k(a x b x c) volume So, if a, b and c are lengths then: a length a x b area a x b x c volume Note that since a x b x c volume then (a x b) x c area x length volume.
Dimensions in Formulae We need to be able to determine the dimensions of any given formula such as the one shown. That is, we have to work out whether it has 1, 2 or 3 dimensions (or is dimensionless). It may be useful for you to have a mental visual picture of the different situations that may occur, so before we start consider the following:
+= a Length b a + b Length - = a b a - b Length a 3 x = a Length x b = In the following diagrams a, b and c represent lengths. AREA ab 3a Length
AREA ab x c Length = AREA ab Volume abc a Length x b = AREA ab Volume abc c Length Division reverses the process and reduces the dimensions. = AREA ab
Volume abc AREA ab = c Length AREA ab b Length = a Volume abc c Length Division reverses the process and reduces the dimensions. = AREA ab
Dimensions in Formulae Example Questions In the formulae below a, b and r represent lengths. State the dimensions in each case. (a) Area (L x L) (b) Length (L + L)(c) Volume (A x L) (d) Volume (V + V)(e) Area (V L) (e) None (V x L)
Dimensions in Formulae Example Questions The table below shows some expressions. The letters x, y and z represent lengths. Decide the dimensions for each expression. ExpressionLengthAreaVolumeNone x + y + z xyz xy + yz + xz ExpressionLengthAreaVolumeNone 2x(y + z) 3xy + z y 3 /x
Stuart tried to derive a formula for the total surface area of a cylinder (including the bottom). He ended up with 3 different formulas only one of which was correct. Which formula is correct and why? r h Can you work out this formula by yourself?