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Area of triangle ABC = | x 3 x 4 Area of triangle = | x base x height
= 6 m2
If, instead, BC is used as the base, then the dashed line in figure c is the height. Since the measurement for the height is not known, the area of the triangle is not readily calculated.
For a solid shape the base is the face of the solid at the bottom. In each of the solids in figures d-f the base is shaded.
(d) (e) (f)
References: Cone, Cylinder, Polygon, Pyramid.
References: Binary Numbers, Exponent, Number Bases.
A bearing is a direction in which to travel. It is an angle measured in a clockwise direction from due north. A bearing is expressed in degrees using three figures, and is used in navigation. Bearings can also be written as compass points.
The arrow in figure a shows a bearing of 270°, which is an angle measured in degrees clockwise from due north. We say that a bearing of 270° is the direction of the arrow in the figure. For bearings which only require two figures, like 45°, a zero is included at the beginning to make it up to the required three figures. For example, a bearing of 45° is written as 045°.
Figure b shows some of the main points of the compass with the equivalent three-figure bearings alongside them. Due north can be expressed as 000°, which is a zero turn, or as 360°, which is a full turn.
Example. There is a lighthouse at the point L in figure c and on a bearing of 150° is a buoy at the point B. The lighthouse keeper visits the buoy, and then sets a bearing from B to travel back to the lighthouse. On what bearing does he travel to get from B to LI
Solution. To solve this problem, we draw a vertical line at the point B to represent due north, because we are taking the bearing from B (see figure d). The next step is to start at B and find the angle turned through from north clockwise in order to travel on the route BL. The fact that the two north lines are parallel enables a geometry theorem to be used:
Angle NBL = 30° Sum of cointerior angles = 180°
The reflex angle NBL = 330° Sum of angles at a point = 360°
The bearing of L from B is 330°.
References: Angles at a Point, Cointerior Angles, Compass Points, Geometry Theorems.
This is a mnemonic for remembering the order of operations when doing calculations with numbers; the order is as follows:
♦ B, Brackets
♦ E, Exponents
♦ D, Division
♦ M, Multiplication
♦ A, Addition
♦ S, Subtraction
BEDMAS is an extremely useful aid in working out problems with complex operations. The instruction to do brackets first means to work out the inside of the brackets before performing the next operation. It is sometimes helpful to insert brackets of your own to further clarify the order of operations. If you enter the numbers and the operations into a scientific calculator in the exact order in which they are written, from left to right, the calculator will automatically perform the operations in the correct order. If there is more than one set of brackets, work from the inside outward. The mnemonic lists division before multiplication, but they have equal ranking. Similarly, addition and subtraction have equal ranking. The following examples show how to obtain answers without using a calculator.
Example 1. Work out (-4 - 2) x -7.
(—4 — 2) x —7 = (—6) x —7 Inside of brackets is done first -6 x -7 = 42
Example 2. Work out 12-7x3 + 8.
12 — 7x3 + 8 = 12 — (7x3)+ 8 Inserting brackets to clarify that
multiplication is done next
= 12-21 + 8
= (12 — 21) + 8 Inserting brackets to clarify the next step
= -9 + 8 = -1
Example 3. Work out 2 - 4 x (-2 - 3)2. Solution. Write
2 - 4 x (-2 - 3)2 = 2 - 4 x (—5)2 = 2 - 4 x 25 = 2 - (4 x 25) = 2-100
Working inside the brackets first Exponents next
Inserting brackets to clarify the next step
Example 4. Work out
4 + 6 2-12
Solution. Division cannot be done first, because the expression is not the same as 4 + 6^2—12, which gives an answer of —5. The expression is the same as (4 + 6) + (2 — 12). Write
4 + 6 (4 + 6)
Inserting brackets to clarify that division is done last
Working inside the brackets first
References: Brackets, Operations.
This is a word used in statistics to describe an unfair sample that has been chosen from a population. A sample is biased if the sample does not accurately represent the population from which it has been chosen. A biased sample cannot be used to make reliable predictions about the population from which it has been drawn. It is necessary, so that the errors in sampling are not repeated, to recognize the reason(s) why a sample is biased and know how to choose another sample that has no bias.
There is a famous story about a poll that was taken by an American magazine just prior to the presidential election in 1936. A huge, 2 million sample was selected
from readers of the magazine, who were then contacted by telephone. The results of the questionnaire predicted a huge defeat for Franklin D. Roosevelt. In reality the opposite happened and Roosevelt was elected by a landslide. The sampling was biased because the sample was restricted only to readers of that particular magazine, and then only to telephone subscribers. A sample can be biased for a variety of reasons. Some reasons are given in the following example.