Unit 2 sec 3.3

Unit 2 sec 3.3 Constructing your own formulas

27 October 2015

23:26

There are many well-known formulas that you can use to solve problems, but sometimes you need to find your own formula. You can construct a formula by following the 3 steps below.

ª    First, identify the subject of formula and the other variables, and units of measurements. This means that you have to decide the purpose of the formula and what the formulae depends on.

ª    Second, find the relationship between the variables. Here, you need to think about how to work out R from d. The length of the return journey R is twice the distance between the 2 places, that is, two lots of d, which can be written as 2d. So the formula is R=2d.

Third, write down all the details of the formula. The formula is R= 2d, where R is the length of the return journey in km, and d is the distance between the 2 points in km. Alternatively, more concisely: the length of the return journey R km is given in terms of the distance d km by the formula R=2d. Not that you should never include units a formula.

Example 12  finding a formula -driving to work

(A)       hearing a working, Anya year drives from home to her office and back 5 times, and she also makes a number of trips from her office to head office and back. Her office is 12 miles from her home in 7 miles from the head office. Find formula for d, where d is the total number of miles that Anya drives in a week when she makes n trip to head office.

(B)         Use your formula to find the distance driven by Anya in week when she makes 3 trips to head office.

Solution

(A)       diagram if it helps you to understand the situation. Then tackle the problem. Step-by-step, by considering separately to return journeys from her home to the office and from the office to the head office.

The distance in miles that Anya drives from her home to her office and that is 12×2=24. So in 5 days, the distance in miles that she drives from her home to her office and back is 5×24=120.

The length of the return journey from the office to their head office is 2×7 miles = 14 miles. So the distance you drive to the head office and back in n trips is in n lots of 14 that is, 14n miles.

The total distance in miles that she drives is the sum of the total distance she travels between home and the office, and the total distance between head office and the office.

Hence the formula for d is

D=120+14,

Where d is the distance in miles, and n is the number of trips from the office to head office.

(B)         Substituting n = 3 into the formula in part (A) gives

D=120+14×3=120+42=162

   Find the relationship between the variables hard to spot, then try some particular numbers first as that might help you to identify the operations involved.

    Then think about which parts of the calculations stay the same and which change. That might help you to discover that the length of the return journey from home to the office is always 120 miles, and that this distance always needs to be added to the distance for the trips to the head office and back.

Example 13, finding a formula - the car ferry

a car ferry can transport both cars and vans stop the van requires a space of 9m, and a car required a space of 5m. find formula for the length L required for c cars and v vans.

Solution

consider the space needed for the current 1^st, try some particular numbers to start with.

One car needs a space of 5m, 30 cars need 2 × 5m = 10m, 3 cars need 3×5m=15m, and so on.

So to find the space needed for c cars, c lots of 5 metres are needed, that is, distance of 5c metres.

Similarly, for the vans.

1 van needs a space of 9m, so 2 vans need 2 × 9m, 3 vans need 3×9m, and so on.

So the space needed for v vans is v × 9 meters, that is 9v meters.

To find the total length, and the distance for the cars and the distance for the vans together.

So a formula for L is L=5c+9v, where L is the total length in meters, c is the number of cars and v is the number of vans.

(As a chair, though, went for, say, 2 cars and 3 barns without using the formula:

the length is 2×5m+3×9=10m+27=37m

if you substitute c=2 and v=3 into the formula, then you obtain the same answer.)

strategy finding formulas

1)   identify the subject of the formula and the other variables, and their units in measurements. If possible, newsletters for the variables that remind you of the context.

2)   Find a relationship between the variables.

3)   Write down all the details of the formula, defining the variables and starting their units (as appropriate).

(It is a good idea to try particular numbers just to suggest what the formula is one, and then check that your formula works.)