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= 20.83 (to 2 dp)
The volume of the tetrahedron is 20.83 cm2, which is | of the volume of the cube.
The regular tetrahedron, which is one of the Platonic solids, has six edges all the same length. This means that each face is an equilateral triangle (see figure c). The apex V of a regular tetrahedron is directly above a point C in the base which is the centroid of the triangular base. The centroid of a triangle is the point of intersection of the medians of the triangle, and is shown in the drawing of the base on the right-hand side of figure c.
The net of a regular tetrahedron is shown in figure d with its base shaded. It is made up of four equilateral triangles, which themselves form a larger equilateral triangle. The area of the surface of a tetrahedron is the same as the area of its net.
Example 2. Find the surface area of a regular tetrahedron if the length of one edge is 6 cm.
TILING PATTERNS 441
Solution. The net of the tetrahedron is drawn in figure e. Each angle of the equilateral triangle is 60°. Write
Area of triangle ABC = \bc sin A See entry Cosine Rule
= |xl2xl2x sin 60° Substituting b = c = 12, angle
A = 60°
= 62.4 (to 1 dp)
The surface area of the tetrahedron is 62.4 cm2.
References: Apex, Cosine Rule, Equilateral Triangle, Median of a Triangle, Net, Platonic Solids, Plane, Polyhedron, Pyramid, Regular Polygon, Vertex.
A theorem is a statement or formula that can be proved by steps of reasoning. The study of geometry as set out by Euclid begins with a few definitions and axioms (statements that we believe are obviously true and do not require proof) and uses them to prove theorems, which in turn are used to prove more theorems, and so on, building a chain. For an example on proving a geometry theorem refer to the entry Proof.
References: Circle Geometry Theorems, Euclid of Alexondria Geometry Theorems, Proof. TILING PATTERNS
442 TIME SERIES
In this entry a time series is taken to be a statistical graph that shows how a quantity varies as time changes. For example, a time series may be a line graph that shows the changes in temperature in New York City during the first 6 months of a given year. Other examples of quantities that may be measured are weight, height, cost, age, popularity, and so on. A time series is a record of what has happened in the past and is used so that a comparison can be made with another time series, or to make a prediction about future changes. A time series may be used in the following way. Suppose you own a supermarket and decide to employ an advertising agency to promote sales. It would be a good idea to first measure sales for a period of 1 year or more and have historical record of how well you are doing. Then bring in the agency and measure sales for the following year for a comparison. In this way you would be able to decide if the agency was effective. There are drawbacks to using historical data to make predictions about the future. You would have to be certain that there were no other influences that could affect your sales. For example, another supermarket may have closed down in your vicinity during the year of the advertising campaign, or there might have been a change in parking restrictions close by and customers suddenly found your supermarket more convenient than before. Whenever time series are used to measure changes and then used in this way for comparisons and predictions great care must be taken to ensure that there is a definite relation between cause and effect.
Example. Ken owns an ice cream company called Ken’s Kones. His accountant provides him with his quarterly sales for spring, summer, fall, and winter over a 3-year period. Based on the following data, draw a time series graph of his sales over this 3-year period, and calculate the moving average on a yearly basis. The order of the moving average is four, because there are four quantities in each yearly cycle,
Sales for Ken’s Kones in $100,000
1 st year 2nd year 3rd year
Spring (Sp) 6.2 7.2 6.2
Summer (Su) 8.1 9.3 10.1
Fall (F) 4.9 6.0 6.5
Winter (W) 3.5 4.2 5.5
Solution. The time series line graph is shown in the figure. The points are joined up to form a series of continuous line segments, but there is no real validity in doing this, because there is no information provided by the accountant about sales from one season to the next. The points are joined up to provide a visual indication of any trends in sales. The average (mean) sales for each year are calculated as follows:
♦ 1st year: (6.2 + 8.1 + 4.9 + 3.5) ^ 4 = 5.7 (to 1 dp)
♦ 2nd year: (7.2 + 9.3 + 6.0 + 4.2) ^ 4 = 6.7
♦ 3rd year: (6.2 + 10.1 + 6.5 + 5.5) ^ 4 = 7.1
Sales in $100000 Ken’s Kones Sales
Three year period
These values are tabulated as follows:
Average Sales per year in $100,000
1st year 2nd year 3rd year
5.7 6.7 7.1
These three “moving averages” 5.7, 6.7, and 7.1 are plotted on the time series and joined with two line segments, as shown on the graph. In this example moving averages are used to represent the trend over the 3 years because there are seasonal variations in the time series that may distract from the overall pattern. We say that the moving averages “smooth” out the seasonal variations.