Dimensions in Formulae - Verulam.S3.Amazonaws
Dimensions in Formulae r
These formulae make sense since they are measures of one, two and three dimensional quantities respectively. So if:
r length
C = 2r
r x r area (length x length)
r
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 = r2
a length
r
a x b area a x b x c volume V
4 3
r
3
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.
r
r2 r
C = 2r
A = r2
r
r2
r3 r
V
4 3
r
3
r3
From the formulae you can see that:
(3) the (1) (2) the length area ofof volume of a the circle a sphere circumference is a is multiple a multiple of the aofcircle the 2 dimensional 3isdimensional a multiple square of cube itsonone on dimensional its radius. radius.
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?
(a ) (b )
(c )
1 3 4 3
2 3
r
C = 2r
2
r
r
r
r
A = r2
r
3
Can you work out this formula by yourself?
V
4 3
r
3
Dimensions in Formulae r
So, if a, b and c are lengths then: a length a x b area
a x b x c volume
C = 2r r
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:
A = r2
ka length
r
k(a x b) area k(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.
V
4 3
r
3
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).
z
4
r
3
2
5a b
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:
In the following diagrams a, b and c represent lengths.
a Length
a Length
3x
a Length
a Length
+ -
b Length
b Length
a+b
=
Length
a-b
=
Length
3a
=
Length
Length
x
b Length
=
AREA ab Length
a Length
AREA ab
Length
b
x
Length
c
x
Length
=
AREA ab Length
=
AREA Volume ab abc
Division reverses the process and reduces the dimensions.
Volume abc
c Length
=
AREA ab
abc c
ab
Division reverses the process and reduces the dimensions.
Volume abc
Volume abc
AREA ab
c Length
Length
=
abc c
abc
c
=
AREA ab
b
AREA ab
=
ab
Length
a
ab b
a
ab
c
Dimensions in Formulae Example Questions
In the formulae below a, b and r represent lengths. State the dimensions in each case. (a ) V (d ) Z =
5ab
4
r
3
2
(b ) L 3 a b 2
7a b
(a) Area (L x L)
(d) Volume (V + V)
(e ) y
ab r
(c ) M 5 r ( a 2 b )
2
(f ) S
(b) Length (L + L)
(e) Area (V L)
3 4
3
r (a 3b )
(c) Volume (A x 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. Expression x+y+z
Length
Area
xy + yz + xz
2x(y + z)
Length
Area
Volume
None
3xy + z
y3/x
None
xyz
Expression
Volume
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?
(a ) 2 r (r
r h
2
2
h )
(b ) 2 r (r h ) 2
(c ) 2 r ( r h )
Can you work out this formula by yourself?
These formulae make sense since they are measures of one, two and three dimensional quantities respectively. So if:
r length
C = 2r
r x r area (length x length)
r
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 = r2
a length
r
a x b area a x b x c volume V
4 3
r
3
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.
r
r2 r
C = 2r
A = r2
r
r2
r3 r
V
4 3
r
3
r3
From the formulae you can see that:
(3) the (1) (2) the length area ofof volume of a the circle a sphere circumference is a is multiple a multiple of the aofcircle the 2 dimensional 3isdimensional a multiple square of cube itsonone on dimensional its radius. radius.
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?
(a ) (b )
(c )
1 3 4 3
2 3
r
C = 2r
2
r
r
r
r
A = r2
r
3
Can you work out this formula by yourself?
V
4 3
r
3
Dimensions in Formulae r
So, if a, b and c are lengths then: a length a x b area
a x b x c volume
C = 2r r
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:
A = r2
ka length
r
k(a x b) area k(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.
V
4 3
r
3
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).
z
4
r
3
2
5a b
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:
In the following diagrams a, b and c represent lengths.
a Length
a Length
3x
a Length
a Length
+ -
b Length
b Length
a+b
=
Length
a-b
=
Length
3a
=
Length
Length
x
b Length
=
AREA ab Length
a Length
AREA ab
Length
b
x
Length
c
x
Length
=
AREA ab Length
=
AREA Volume ab abc
Division reverses the process and reduces the dimensions.
Volume abc
c Length
=
AREA ab
abc c
ab
Division reverses the process and reduces the dimensions.
Volume abc
Volume abc
AREA ab
c Length
Length
=
abc c
abc
c
=
AREA ab
b
AREA ab
=
ab
Length
a
ab b
a
ab
c
Dimensions in Formulae Example Questions
In the formulae below a, b and r represent lengths. State the dimensions in each case. (a ) V (d ) Z =
5ab
4
r
3
2
(b ) L 3 a b 2
7a b
(a) Area (L x L)
(d) Volume (V + V)
(e ) y
ab r
(c ) M 5 r ( a 2 b )
2
(f ) S
(b) Length (L + L)
(e) Area (V L)
3 4
3
r (a 3b )
(c) Volume (A x 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. Expression x+y+z
Length
Area
xy + yz + xz
2x(y + z)
Length
Area
Volume
None
3xy + z
y3/x
None
xyz
Expression
Volume
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?
(a ) 2 r (r
r h
2
2
h )
(b ) 2 r (r h ) 2
(c ) 2 r ( r h )
Can you work out this formula by yourself?
