International projectionist (Oct 1931-Sept 1933)

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November 1931 INTERNATIONAL PROJECTIONIST 25 Figure 3 Z -2 I-! 8 SQUA /NC fSE H£S -4 decreased from 10 to 5, or 2 to 1. Yet the amperes increased in the proportion of 1 to 2. This is an example of "inverse" ratio and proportion. This is not the same as "direct" ratio and proproportion. For example: Problem: If 5 cents buys 1 apple, how many apples can be bought for 15 cents? 5:1 X X 5X X 15 :X 15 X 1 15 3 apples, Answer. The foregoing is an example of "direct" ratio and proportion. Where the number increases directly as the price increases, it will be seen that we compare the price of one article with the price of X articles. In solving any problem like the foregoing, it is always good policy to ask oneself the following: "A is to B as how many is to C?" Or: A:B = X:C. Or: B (No. 3) (No. 4) In solving equation No. 3 we get: B X X = X(AXC) Or: BX = AC If we divide both sides of this equation by B, we get: BX B AC B AC Or: X = (since B cancels on the B left side) In solving the same proportion by , A X means of equation No. 4 (ff) C a simple method is available. By crossmultiplying we obtain the same result. By this is meant: draw a line from B to X and multiply them together. Draw an Figure 2 other line from A to C, multiply them together, and let the product of the first two letters equal the product of the second two letters. Thus: Figure 5 B xX = A X C Or: B X = AC AC X = (dividing both sides by B), B Answer. Plane Geometry The Square. The areas of various geometrical figures may be determined by employing simple formulae. For example: a square is said to be a figure the length of which equals its width. Therefore, a figure having a length of 1 inch and a width of 1 inch is said to be a one 4 ARE INC HES. ■ ' U 2'^ J 9 3' S QUA RE 4CHES 1 , ' f h 3" ^ UNIT C/M(TS Figure 4 inch square! In mathematics we call this area one square inch. Suppose we extend the length of this square to 2 inches, and also make the width 2 inches. Would we now have 2 square inches? We would have 4 square inches. If the length be 3 inches, and the width 3 inches, we have an area of 9 square inches. (Fig. 2.) The conclusion of this is that the area of a square is obtained by multiplying the length by the width. The Rectangle The same that is true for the square is true for the rectangle. A rectangle really is made up from a number of squares. If UNITS the length of a rectangle is 2 inches and its width 1 inch, the area is 2 square inches. If the length is 4 inches and the width is 2 inches, the area is 8 square inches. In other words, the area of a rectangle is equal to the product of its length multiplied by the width. (Fig. 3.) The Triangle The area of a triangle can be computed in a similar manner. Let us consider a square the length of which is 2 inches and the width of which is 2 inches. The area of this square is 4 square inches. Now let us draw a diagonal line within the square so as to divide the square into two triangles having equal areas. (Fig. 4.) Since the sum of two rectangles is equal to the entire area, then the area of each triangle is one-half the area of the square. Each of these triangles differs from the square in a certain respect. It appears as though a triangle is a square having one of its sides squeezed down to a point. Therefore, the width of the triangle is not 2 inches all the way through. Its width varies from 2 to zero. Its average width is 1 inch. The area of a triangle, then, is measured by multiplying its length by its "average width." A triangle is said to have an altitude and a base. These correspond to the lengths and the widths of squares. The area of a triangle, then, is the product of the "average base" multiplied by the height or altitude. Expressed by equation: Area of triangle = % b h . . (No. 5) where: b = base h = height or altitude In the above figures, then, the area of either triangle is 2 square inches, for: A = 1/2 X 2 X 2 = 1/2X4 = 2 square inches. Answer. The Circle The area of a circle depends upon its diameter. If we increase the diameter, the area increases. Therefore, if we know the diameter or the radius (which is half the diameter), we may determine the area. Let us see how the area of a circle increases with the diameter. Suppose a circle has a diameter of 1 unit and its area is 1 square inch. If we double the diameter, the area does not double. If the diameter becomes twice as great,