Formulas


Introduction


A formula is just a statement about the relationship between various quantities.

Example

P = mv

is a formula encountered in Physics - it says that the momentum (P) of a body is equal to the mass (m) of the body multiplied by its velocity (v).

Example

Another physics formula is

\[ E = \frac{1}{2} m v^2 \]

which says that the kinetic energy (E) of a body is equal to half multiplied by its mass multiplied by its velocity squared ('velocity squared' means v × v - note that it is only v that being squared in this formula).

Example

The area of a rectangle is given by multiplying its length by its breadth. Stated as a formula, if A is area, l is the length and w the breadth (or width), then

A = l × w,     or     A = lw

If the length and breadth were identical, then we would have a square, and the above formula would reduce to

A = d2

where d is the length of each side (compare this with the equation

\[ E = \frac{1}{2} mv^2 \]

where we descibed the last term as 'v squared'. Hopefully you can now see the derivation of this phrase)

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Lines, Areas, Volumes


The area of a rectangle (or square) was mentioned in an example above. This concept of an area being given by two lengths multiplied together generalizes to other expressions. So

p r2

could be an expression for area because it has two lengths multiplied together, and in fact it is the area of a circle of radius r.

However an expression like

2 p r

could not be a formula for an area because it does not contain two lengths multiplied together.

Quick Quiz


Which of the following could be a formula for an area (stressing the word could - we are not guaranteeing that it is a valid formula for a definite shape. At this stage, we are just eliminating those formulas that are definitely not area formulas)

  1. abc
  2. a2
  3. ab2
  4. 3ab
  5. 6ab
  6. 7a2



Following the same logic

  • A formula consisting of three lengths multiplied together could represent a volume. So if we are told that

    \[ \frac{4}{3} \pi r^3 \]

    is a valid geometrical formula, then we know straightaway - without being given any more information - that it will represent a volume.

  • Going the other way, a formula for the length of a line will tend to contain just one length (which is not squared or raised to any power). For example

    2 p r

    which has previously been mentioned as the length of the circumference of a circle.

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Perimeters

The formula for the perimeter of a shape is straightforward - it is just the addition of all the separate sides of the shape. So for a rectangle, the perimeter (P) would be

P = l + l + w + w

    = 2l + 2w

which also equals

P = 2 ( l + w )

if you know about factorization.

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Standard Formulas


Just a few more formulas that are quite common

Area of a Triangle

\[ \frac{1}{2} \times \ \mbox{base} \times \ \mbox{height} \]

which here would be

\[ \frac{1}{2} \times a \times h \]

\[ = \frac{ah}{2} \]

Straight Line

y = mx + c

which here would be

y = 2x + 1

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Circles



The two most important formulas for circles are

Circumference = 2 p r   
( or alternatively Circumference = p d,   where d is the diameter)

Area = p r2


Differentiate between these two formulas by remembering what we have said previously about recognizing the form of line or area formulas. So of the two expressions above

p r2

must represent the expression for the area of the circle.

For completeness, when you come to find the volume of a sphere, you might expect that the formula will contain three lengths (or variable terms) multiplied together. And this is indeed so : The formula for the volume of a sphere is

\[ \frac{4}{3} \pi r^3 \]

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Past Exam Questions


Two parents are making a round sandpit for the children. It has a radius of 1.5 metres and a depth of 0.2 metres.


1. What is the volume of sand needed to fill the sandpit?

  • A     0.18m3
  • B     1.35m3
  • C     1.80m3
  • D     6.75m3

2. Coloured warning tape will be put all the way round the top edge of the sandpit. How much tape will be needed?

  • A     4.5m
  • B     6m
  • C     6.75m
  • D     9m



Stage 3

Jane travels to work by car. Each year she travels 40 miles a day, for 230 days. She is considering buying a new car. This is the information she has:


She calculates next year's costs for each type of car, excluding insurance and maintenance, using the expression:

(cost to buy) + (fuel cost per mile $\times$ number of miles)

For the diesel car D, this gives the equation: D = 13 500 + 0.08M where M is the number of miles travelled.

a)   Write down a similar equation for the petrol car, P, using the information in the table.

b)   Find the value of D for next year.

c)   Find the value of P for next year.

d)   How many miles would Jane have to travel next year for P and D to have the same value?

e)   Describe briefly how you checked that the value of M in part (d) is correct.


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