SECTION XV BUSINESS MEASUREMENTS Finding Areas In finding the number of square inches in a rectangle 5 inches by 3 inches the child naturally divides it into 1-inch squares and ascertains their number by counting. In a short time he ascertains that the number of square units into which a rectangle can be divided is equivalent to the number of square units in one row multiplied by the number of rows. At length he formulates the rule: with the understanding that the dimensions are expressed in the same linear units, and that their product gives the area in the corresponding square units. Primitive Method The early surveyor, whose schooling was limited, developed a method of ascertaining with sufficient accuracy (when land was cheap) the area of an irregular field, such as ABCD. On a piece of paper fastened to a horizontal board placed at A, for instance, he drew a line in the direction of AB and another in the direction of AD, thus obtaining the angle at A. He then measured AB. At B, C, and D, he drew the angles; he also measured the sides BC, CD, and DA. From these data he made a drawing of the plot to a convenient scale, say, 1 inch to 10 rods. He next divided the drawing into 1-inch squares, counted the whole squares, made due allowance for the fractions, and multiplied 100 square rods by the total number of squares. Division into Triangles After he discovered the fact that only one triangle could be made with sides of a given length, the necessity of mapping the corner angles disappeared when the diagonal DB could be measured as well as the other sides. Later the surveyor found that he could ascertain the area of the given field by measuring DB, the common base of the two triangles, and AX and 40 100 Y B 50 CY, their respective altitudes, each triangle being one half a rectangle of the same dimensions. While these three lengths would be sufficient to ascertain the area of the field, they would not determine its shape. For this purpose the points X and Y must be located by measuring, say, DX and DY. Drawing to Scale A person is helped in the solution of many problems by the employment of a rough diagram which suggests the required operations. In some cases, however, a drawing must reproduce accurately the proportionate length of each line therein. To enable a person using a map to determine the distance between any two places, a scale is furnished, from which the equivalent in miles can be ascertained. Map scales vary according to the extent of country represented. A section 300 miles long, 200 miles wide, on a map 6 inches long, 4 inches wide, might be accompanied by a scale 2 inches long to represent 100 miles, with 10-mile subdivisions. The statement that the map was drawn on a scale of 50 miles to 1 inch would, however, be sufficient for most purposes. Sight Exercises 1. On a scale of 1 inch to 40 rods, how long a line is required to represent : a. 85 rods? b. 75 rods? e. 40 rods? f. 50 rods? c. 125 rods? d. 140 rods? g. 150 rods? h. 130 rods? Written Exercises 1. On a scale of 1 inch to 40 rods draw a line DB representing 140 rods. On this line mark a point X, 40 rods from D, and Y, 90 rods from D. At X erect a perpendicular representing 75 rods and note its upper extremity by the letter A. At Y let fall a perpendicular representing 120 rods and note its extremity by C. Draw AD, AB, BC, and DC. (See diagram page 415.) 2. (a) How long is AD? (6) How many rods does it represent? (c) How long is AB? (d) How many rods does it represent? (e) How long is BC? (f) How many rods does it represent? (g) How long is CD? (h) How many rods does it represent? 3. Calculate the length (a) of AB, when AX is 75 rods and BX is 100 rods. (b) Of BC, when BY is 50 rods and YC is 120 rods. (c) Of CD, when CY is 120 rods and YD is 90 rods. (d) Of DA, when DX is 40 rods and AX is 75 rods. Angles When two intersecting lines make four equal angles, the lines are said to be perpendicular to each other and each of the angles is called a right angle. Angles are measured in degrees, the size of the angles depending on the portion of the circumference of a circle embraced between the lines forming the angle when the intersection of the lines is at the center of the circle. As the circumference of a circle contains 360°, a right angle contains 90°. 90 90° 90 90 When two lines intersect obliquely, two of the angles are smaller than right angles and two of them are larger. An angle smaller than a right angle is called an acute angle; one larger than a right angle is called an obtuse angle. Triangles Considering the length of their sides, triangles are equilateral, having three equal sides; isosceles, having two equal sides; scalene, having unequal sides. The three angles of an equilateral triangle are equal, each containing 60°. The angles opposite the equal sides of an isosceles triangle are equal, the side opposite the third angle being called the base, regardless of its position. A triangle containing a right angle is called a right triangle; one containing an obtuse angle is called an obtuse-angled triangle; one containing three acute angles is called an acute-angled triangle. Quadrilaterals A quadrilateral is any figure of four sides. When the opposite sides are parallel, it becomes a parallelogram. The opposite sides of a parallelogram are equal, as are the opposite angles. When the angles of a parallelogram are equal, the latter becomes a rectangle. A square is an equilateral rectangle. Area of a Rectangle To find the area of a rectangle is to ascertain the number of square units it contains; square inches, square feet, square miles, etc. When the rectangle ABCD is 9 units long and 4 units wide, the number of square units will consist of 4 rows of 9 square units each, or 9 rows of 4 square units each. A C B The number of square units in the area of a rectangle is equal to the product of the number of linear units in its length by the corresponding number in its width. Mathematicians frequently denote one side of a rectangle as its base and an adjacent side as its perpendicular. In formulating the rule for finding its area, they give it thus: Area of rectangle Base x Perpendicular Written Problems 1. How many acres are there in a rectangular field 125 rods long and 84 rods wide? 2. Find the cost of painting a roof 24 feet wide and 48 feet long at 75 cents per square yard. |