Unit 2 sec 3.1

Using formulas Unit 2 sec 3.1 from words to letters

23 October 2015

22:32

   Solving a problem in mathematics often involves using a formula. I used extensively in everyday life.

   Although some formulas are easy to remember when expressed in words most formulas are written in a more concise form. This is particularly true when they are used in computer programs or spreadsheets, or when they are more complicated.

   Many things are represented by a symbol; for example the symbol like a T on a road sign, the symbol P on a map often indicates a car park. Symbols are concerns - they say writing out a whole word or sentence and consequently they make it possible to see information more clearly. In mathematics, you are already familiar with some symbol, such as ÷ and √

   used to estimate the distance travelled by car. If we use the letters

s to represent the average speed,

t to represent the time taken,

d to represent the distance travel.

Then this word formula can be written more concisely as the “letter formula”

  D= s×t.

The letters are used instead of words in a formula, then it is essential to say what quantities the letters represent.

Using a formula

   The letters in a formula stands for numbers that are related in the way given by the formula. Thus you can think of a formula as a way of summarising the calculation process.

  When you use a formula, you replace the letters to the right of the equals sign (in the case, s and t) by numbers, and then carry out the calculation to find the value of the letter to the left (in this case, d).

   This process is known as substituting values into the formula.

Rather than using the word formula to find the distance, breaking news the more concise formula d=s×t, where d, s and t are defined as before.

  Replacing s by 50 and t by 1.2 gives

D= 50×0.2 = 60 hence the distance travelled is 60km.

    Since the values of s, t and d can vary and represent different numbers in different scenarios, they are known as variables. In general, and letter can represent different numbers is called a variable.

So formula is an equation in which one variable, called the subject of the formula, appears by itself on the left-hand side of the equation and only the other variables appear on the right-hand side. Thus, enables you to calculate the value of the subject when you know the values of the other variable stop

   In many formulas the variables represent measurements, and it is important to check that the values you substitute are measured in appropriate units.

Formula with set units

   Some formulas the units are already set and cannot be changed. No other units can be used in this formula. Before you substitute in a formula like this, you must check that the information that you use is express in the correct units, and make any conversions.

Examples 7 substituting value into a formula

  a European car hire company charges €50 per day for that higher of the small car, plus a booking fee of €20. So, the total cost of hiring the car is given by the formula

T= 50×n +20

Where T is the total cost in € and n is the number of days for which the car is hired.

How much does it cost to hire the car for 2 weeks?

Solution

check that the given information is in the correct units.

In formula, the time n is measured in days, so 1^st convert to weeks in 2 days.

There are 7 days in one week, so in 2 weeks there are 2× 7 days= 14 days hence n =14

                

 substitute in do the calculation

substituting n = 14 into the formula gives

t = 50× 14+ 20

  = 700+ 20

  = 720

State the conclusion, including the correct units.

Hence the cost of hiring the car for 2 weeks is €720.

Once you have substituted numbers into formula, you perform the calculation by using the usual rules of arithmetic. The mnemonic BIDMAS helps you to remember the order of operations.

Brackets, then indices (powers and roots), then divisions and multiplications, then additions and subtractions.

Example 8 calculating the time to walk uphill

  Naismith’s rule estimates that the time taken for a walk uphill is given by the formula

T= +

Where

T is the time for the war in hours,

D is the horizontal distance walked in kilometres,

H if the height climbed in metres.

(A)       estimate how long a walk will take the horizontal distance is 20km and their height is 1200m

(B)         why might you need to allow longer than this estimate?

Solution

(A)       check that the given information is the correcting units.

In this case, the horizontal distance is 20km and climbed is 1200m. The units here are those specified for the formula, so no conversion is needed.

Checked and do the calculation.

Substituting D =20 and H = 1200 into the formula gives

T=  +  = 4+2=6

State the conclusion, including the correct units.

Hence Naismith’s rule predicts a 6 hour walk.

(B)         you may need to allow longer than 6 hours to accommodate to rest breaks, all because the terrain is difficult, the walkers are on the or the weather is bad.

Formulas you can choose the units

The formula for the area of a rectangle is A= l×w, where a is the area l is the length and W is the width. You can choose which units to use as long as these units are consistent. The area in square centimetres; if you measure the length in kilometres, then the width should also be measured in kilometres.

   The units given are not consistent with each other, then you should convert the measurements into appropriate units before substituting them into the formula.

In formula d= s×t, mentioned earlier, the units could be consistent; the units of the speed are km/h, then the unit for the time are h and the unit for the distance are km

Example 9 substituting values into a formula

a car travels at an average speed of 95km/h. Use the formula

d= s×t

defying the distance the car travelled in each of the following signs.

(A)       2.5 hours

(B)         40 minutes

give your answers to 2 significant figures

Solution

(A)       unit for speed is km/h and the unit 4 time hours, then the distance in km, so no conversions are required.

When s= 95 and t= 2.5,

D=95×2.5=237.5

so the distance travelled is 240km (to 2 s.f.).

(B)         first, converted the given time into hours. Since

40 min = h = 0.666h,

the time is 0.666h

when s = 95 and t = 0.666

d= 95×0.666= 63.33

so the distance travelled is 63km (to 2 s.f.).

The consistency of units is to substitute the values for the variables together with the units into the formula. The area of the a rectangle in length 3m and width 50cm could have been written as

A= 3m×0.5m

   = (3×0.5)m²

   = 1.5m²