This continued fraction' is reduced to an ordinary fraction by commencing at the lower extremity thus: But the great use of this method is not in enabling us to obtain the exact measure of the ratio CD : AB, but an approximate value, which shall be as near as we please to the true value. Thus, in this case, 3 is an approximate value; 3+, or 34, is a nearer value; and or 3, is nearer still. And similarly, whatever be the length of the continued fraction, by stopping at any particular quotient, and neglecting the remainder, an approximate value of the fraction is obtained, which differs less and less from the true value according as more of the continued fraction is taken into account. And it will be seen in the above case, (as it is indeed in all others,) that the approximate values 3, 3, 31, taken in order, are alternately less and greater than the true value 33. The last approximate value, 31, differs from the true value, 33, by only one twenty-fourth of the unit. PROB. 16. By a similar method to that employed in the last Prob., to compare two given angles with each other, or to find the measure of a proposed angle in terms of some given unit. Any one of the following ways may be employed to measure the angle ACB:— (1) With centre C, and greatest radius that can be conveniently used, describe an arc of a circle not less than the sixth part of the whole circumference, and cutting the two lines which bound the given angle in A and B. Then with centre A, and the same radius as before, describe another arc intersecting the former in D; AD* is an arc of 60o. B D Then, since in the same circle arcs are proportional to their chords, by stepping AB, with the compasses, along AD, as described in Prob. 15 for straight lines, the arc AB may be compared with the arc AD, just as the straight line AB, in the former case, was compared with the straight line CD. It is usual to speak of an arc of so many degrees, meaning an arc which subtends that angle at the centre. [In these Exercises is taken to be 3·1416, when it is not otherwise stated.] (1) Find the area of an isosceles triangle, in which the angle contained by the equal sides is 120o, and the altitude of the triangle is 6 ft. 3 in. Ans. 67-656 sq. ft. (2) Two adjacent sides of a triangle measure 356, and 44-2, yards, and contain an angle of 30°; find the area of the triangle. Ans. 393-38 sq. yds. (3) Having a given circle traced out before you, shew how to trace another, whose circumference shall be exactly 4 times that of the former. How would you trace one whose area shall be 4 times that of the first? (4) The circumference of a circle is 38 inches; find the length of a side of the greatest equilateral triangle which can be cut out of it. Ans. 10 475 in. (5) The radii of two circles are 5, and 12; find the radius of another circle, whose area shall be exactly equal to the sum of the areas of the other two. Ans. 13. (6) In cutting out the greatest square from a given circular board, how much of the material is wasted? (T = 22). 4 Ans. or a little more than one-third, of the whole. 11' (7) In cutting out the greatest equilateral triangle from a given circular board, how much of the material 22 is wasted? (~=27). Ans. 586, or more than half, of the whole. (8) The diagonal of a square is 45 yards; find the area of the inscribed circle. Ans. 795 2175 sq. yds. (9) Compare the area of a square with the sum of the semi-circles described upon its sides. Ans. 2 T. (10) Two radii of a circle are at right angles to each other, and a chord is drawn joining their extreme points; compare the segments into which the circle is thus divided. (T = 22). Ans. 10: 1. (11) If the area of a circle be 16 sq. yds., find the area of a sector of the circle, whose arc subtends at the centre an angle of 75o. Ans. 3 sq. yds. (12) The radius of a circle is 25 feet, and the angle of a sector of it contains 63°; find the length of the arc, and thence the area of the sector. (1) Ans. 27 ft. nearly. (13) Find the length of an diameter of the circle being 6 feet. (2) Ans. 343 sq. ft. arc of 17° 10′, the Ans. 8988...... ft. (14) A portion of wood cut by a circular saw shews an arc of a circle on its face made by the teeth of the saw, of which the chord is 9 inches; and a perpendicular from the arc to the middle of the chord is 1.35 in.; find the diameter of the saw. Ans. 16.35 in. (15) An acre of ground, in the form of a circle, has a walk cut from it all round, 2 yds. broad, and the rest is grass; find the radius of the original circle, and the area of the grass-plot. (1) Ans. 39.25 yds. (2) Ans. 4359 1584 sq. yds. (16) The areas of two concentric circles are 165 yds. and 132 yds. respectively; find the breadth of the annulus between the circumferences. Ans. 2.3 ft. nearly. (17) It is required to construct, exactly in the middle of a circular area of 7 acres, a circular pond, which shall occupy one-third of the whole ground; find the radius of the pond, and the width of the ground left. (1) Ans. 59.94 yds. (2) Ans. 43.89 yds. (18) Find the area of the annulus formed by the super-position of a circle, whose diameter is 26.5 feet, on another circle whose diameter is 28.2 feet. (19) An animal, tethered by a rope fastened to a stake in the straight hedge of a field, is allowed an acre of grass; what will be the length of the rope? Ans. 555 yds. nearly. (20) Two circles, each having a radius of 1 inch, intersect so that the circumference of each passes through the centre of the other; find the area which is common to both. Ans. 1.228 sq. in. nearly. (21) Two circles touch one another internally, the radius of the larger circle being 2 in.; find the distance between the centres, when the area of the smaller circle is half that of the larger. Ans. 5858 in. (22) Two circles touch one another externally, the areas being as 23: 1; find the distance between the centres, if the smaller radius be 1 inch. Ans. 23 in. (23) Find the measure of the angle at the centre of a circle which is subtended by an arc equal to the diameter. Ans. 114° 35', nearly. |