unit 2 sec 3.2

Unit 2 sec 3.2 writing formulas concisely

27 October 2015

22:25

  As containing a lot of mathematical symbols can look quite complicated. To make them more concise, change multiplication are usually omitted.

   However, when you substituting numerical values into a formula, you usually have to part the change multiplication sign back in, to make the meaning clear. So 3 × y can be written as 3y, the 3 × 2 cannot be written as 32. One way to check that you understand what the given Formula means is to try describing in words how to use the formula.

Conventions for writing formulas

these are several conventions that are usually followed when writing formulas concisely stop

¨    in products, numbers are usually written 1^st; for example, the formula k= 1.6 × m, or equivalently K = M × 1.6, is erecting concisely as k= 1.6m.

similarly, (2a+b) ×3 is written concisely as 3(2a+b). However (2a+b)c and c(2a+b) are both acceptable ways of writing (2a+b) ×c.

¨    In products, letters are often written in alphabetical order; for example, d= s×t, or equivalently d=t×s, is usually written as d= st

¨    finally, divisions are usually written in the form of a fraction; for example s= d÷t is written as s=  and read as ‘s equals d over t’.

     This helps to distinguish between say, the distance 5 metres, which is printed as 5 m and the expression 5 × m (that is, 5 × the variable m), which is printed concisely 5m. These work identical and the meaning is obtained from the context. This is one reason why units are usually not included in mathematical calculations that involve variables.

Example 10 estimating the volume of a log

foresters can estimate the binding of a log of wood by using the formula. V=  

where V is the volume of the log ins cubic metres, L is the length of the log in metres, D is the distance around the middle of the log in metres, and Ԉ is approximately 3.14159.

Estimate the volume of the log that is 1.5 m long and 92 cm around the middle, giving your answer to 2 significant figures.

Solution

check that the given information is in the correct units.

The length is 1.5 m, so L= 1.5. The distance around the middle is 92 cm, but the formulae requires the measurement in metres. Since 92 cm = (92÷ 100) m =0.92, we have D= 0.92.

Substitute and do the calculation.

Substituting, L= 1.5 and D=0.92 into the formula V =  gives

V =  =  = 0.101

date the conclusion, including the correct units. Hence the volume of the log is 0.10m³ (to 2 s.f).

     Several differing calculator sequences can be used to calculate the final answer, and some of these sequences involved using the memory and other function keys on your calculator.

Substituting negative numbers

when you replace a letter by a negative number, it is usually helpful to include the number in brackets to avoid confusion.

Example 11 substituting a negative number

consider the formula A= c² - 5c + 3. Find the value of A when

c = -2.

Solution

potting brackets around -2 and substituting it for c gives

A= (-2) ² - 5(-2) +3

   = (-2) × (-2) –(-10) +3

   = 4÷10÷3

   =17.

-2 has been enclosed in brackets when it is substituted to ensure that the - is not separated from the buying mistake.

  Then remember that the calculation above – (-10) means subcontract -10, and subtracting the negative number -10 is the same as adding the corresponding positive number 10. So in this calculation, -(-10) is the same as + 10.

However in the variable being substituted appears 1^st in the calculation on its own, then no brackets are required. For example, if A= C+ 3 and C= -2, then A= -2+3=1.

Sometimes when you are substituting into a formula you have to find the negative of a number. This is the number that is produced by putting a - in front of the number.