(2) The circumference of a circle is 360 feet. 314163600011459 31416 45840 31416 144240 125664 185760 157080 286800 282744 4056 Thus the diameter is about 114.59 feet. 113. We will now solve some exercises which depend on the Rules already given. (1) Find the diameter of a carriage wheel which is turned round 1000 times in travelling a mile. Here 1000 times the circumference of the wheel is equal to 1760 yards; thus the circumference is 1.76 yards. Then, by Art. 111, the diameter is 7×08 yards, that is, 56 of a yard. 7 122 × 1·76 yards, that is, (2) Suppose that the distance of the earth from the sun is about 95000000 miles, and that the earth describes a circle round the sun in 365 days: find the number of miles described by the earth in one minute. The circumference of the circle described by the earth is about 2 × 95000000 × 3·1416 miles, that is, about 596904000 miles. In 365 days there are 525960 minutes. Divide the number of miles by the number of minutes; thus we obtain very nearly 1135 miles. EXAMPLES. VIII. Assuming that the circumference of a circle is 3 times the diameter, find the circumferences of the circles with the following diameters: 1. 14 feet. 3. 213 yards 2 feet 8 inches. 4. 1 furlong 60 yards. 2. 86 yards 1 foot. Assuming that the circumference of a circle is 3:1416 times the diameter, find the circumferences of the circles with the following diameters: 5. 27 feet. 6. 61 yards 2 feet. 7. 555 yards 1 foot 6 inches. 8. 1 furlong 80 yards. Assuming that the circumference of a circle is 3 times the diameter, find the diameters of the circles with the following circumferences: 9. 66 yards. 11. 3 furlongs 4 chains. 10. 10 chains. 12. 1 mile. Assuming that the circumference of a circle is 31416 times the diameter, find the diameters of the circles with the following circumferences: 13. 1 foot. 15. 108 yards 1 foot. 17. Suppose that the planet Mercury describes in 88 days a circle round the Sun of 37000000 miles' radius: find the number of miles described by the planet in one second. 18. The diameter of a carriage wheel is 28 inches: find how many turns the wheel makes in travelling half a mile. 19. A road runs round a circular shrubbery; the outer circumference is 600 feet, and the inner is 480 feet: find the breadth of the road. 20. The difference between the diameter and the circumference of a circle is 10 feet: find the diameter. T. M. 4 IX. ARC OF A CIRCLE. B 114. Let C be the centre of a circle, AB any arc of the circle, AD a quarter of the circumference. The length of AB is to the length of AD in the same proportion as the angle ACB is to the angle ACD, that is, as the angle ACB is to a right angle. Therefore the length of AB is to the circumference of the circle in the same proportion as the angle ACB is to four right angles. 115. Angles are usually expressed in degrees, 90 of which make a right angle; and consequently in four right angles there are 360 degrees. A degree is subdivided into 60 minutes, and a minute into 60 seconds. 116. Symbols are used as abbreviations of the words degrees, minutes and seconds. Thus 6° 23′ 47′′ is used to denote 6 degrees, 23 minutes, 47 seconds. 117. The number of degrees in the angle subtended by an arc of a circle at the centre being given, to find the length of the arc. RULE. As 360 is to the number of degrees in the angle, so is the circumference of the circle to the length of the arc. 118. Examples. (1) The circumference of a circle is 48 inches, and the angle subtended by the arc at the centre is 54 degrees. 360: 54: 48: the required length, Thus the length of the arc is 7.2 inches. (2) The circumference of a circle is 25000 miles, and the angle subtended by the arc at the centre is one degree. 360 1:25000: the required length, 36) 2500 (694 216 340 324 160 144 16 Thus the length of the arc is about 69.4 miles. 119. The length of an arc of a circle being given, to find the number of degrees in the angle subtended by the arc at the centre of the circle. RULE. As the circumference of the circle is to the length of the arc, so is 360 to the number of degrees in the angle. 120. Examples. (1) The circumference of a circle is 50 feet, and the arc is 8 feet. 50 8 360 the required number of degrees, (2) The circumference of a circle is 25000 miles, and the arc is 750 miles. 121. The chord of an arc being known, and also the chord of half the arc, to find the length of the arc. RULE. From eight times the chord of half the arc subtract the chord of the whole arc, and divide the remainder by three. This Rule is not exact; it gives the length of the arc smaller than it ought to be. If the arc subtend at the centre of the circle an angle of 45 degrees, the error is about of the length of the arc: the error increases rapidly as the angle increases, and diminishes rapidly as the angle diminishes. 1 20000 122. Examples. (1) The chord of an arc is 14 inches, and the radius of the circle is 25 inches. By Art. 94 the chord of half the arc is about 7·0710678 inches. 7.0710678 8 5 6 5 6 8 5424 14 3 425 685424 1 41 895 141 Thus we obtain for the length of the arc 14:1895141 inches. (2) The chord of an arc is 58 inches, and the radius of the circle is 100 inches. By Art. 94 the chord of half the arc is about 29.32 inches. 29.32 23456 58 317656 5.8.85 Thus we obtain for the length of the arc 58.85 inches. |