3.02 Substitution into formulas
Using algebra we can write relationships between different quantities concisely in formulas.
For example, instead of writing, "The area of a rectangle is calculated by multiplying the length by the width," we can write the algebraic formula $A=LW$A=LW. Formulae give a general rule relating variables and we can use substitution to evaluate the formula for specific cases, just as we substituted particular values for length and width when finding areas of rectangles in the measurement chapter.
There are lots of topics that have common formulae: simple interest, perimeter, area, volume, temperature conversions, and many more across a broad range of areas in Mathematics, Physics, Chemistry, Finance, and so on. It is a good idea to be familiar with any formulae that are related to a given topic before working through exercises for that topic.
Worked example
The volume of a square based pyramid is given by the formula $V=\frac{1}{3}b^2h$V=13b2h, where $b$b is the length of a side of the base square and $h$h is the perpendicular height.
Find the volume of a square based pyramid with $b=6$b=6 m and $h=5$h=5 m.
Think: Write out the formula, substitute in the known values, and then evaluate the resulting expression.
Do:
| $V$V | $=$= | $\frac{1}{3}b^2h$13b2h |
Write out the formula. |
| $=$= | $\frac{1}{3}\times6^2\times5$13×62×5 m3 |
Replace $b$b with $6$6 and $h$h with $5$5. |
|
| $=$= | $\frac{1}{3}\times36\times5$13×36×5 m3 |
Evaluate the term with the power and then multiply. |
|
| $=$= | $60$60 m3 |
Evaluate, and don't forget units for questions in context. |
Practice questions
question 1
The perimeter of a square with side lengths of $a$a is given by the formula $P=4\times a$P=4×a.
Find $P$P if the length of each side is $5$5 cm.
question 2
The area of a triangle is given by the formula $A=\frac{1}{2}$A=12$($(base$\times$×height$)$).
If the base of a triangle is $3$3 cm and its height is $10$10 cm, find its area.
Question 3
The simple interest generated by an investment is given by the formula $I=\frac{P\times R\times T}{100}$I=P×R×T100.
Given that $P=1000$P=1000, $R=6$R=6 and $T=7$T=7, find the interest generated.
Question 4
Energy can be measured in many forms. A quantity of energy is given in units of Joules (J).
The kinetic energy, $E$E, of an object in motion is calculated using the following formula:
$E=\frac{mv^2}{2}$E=mv22
where $m$m is the mass of the object in kilograms and $v$v is the speed of the object in metres per second.
Find the kinetic energy, $E$E, of an object with a mass of $6$6 kg, travelling at a speed of $19$19 metres per second.