= 10 feet, 2. The line A B = 30 feet, B C 20 feet, CD 18 feet; find the lengths of AD = 8 feet, and ab Section 2. 3. The length of the shadow of St. Paul's is 52 yards, when the shadow of the Monument is 24 yards; what will be the length of the shadow of the Monument when the shadow of St. Paul's is 62 yards? 4. The length and breadth of a town are 31, 24 miles respectively; the length on a plan is 4 feet 5 inches; what is the breadth ? 5. The circumference of a circle, whose diameter is a unit, is 3 1416; find the circumference of another circle whose diameter is 49. Section 3. 6. What is the circumference of a rope whose diameter is 3} inches? 7. Find the circumference of an iron bolt whose diameter is 1 inch. 8. The circumference of an iron bolt is 6 feet 9 inches; what it its diameter ? 9. The circumference of an iron axle is 4 feet 8 inches; find its diameter. Section 4. 10. The length of a shadow of any object is 84 feet, while the shadow of a 3 feet cane is 2 feet 6 inches; what is the height of the object? 11. The circumference of a barrel is 24 feet 1 inch; find its diameter. 12. The circumference of a circle is 116 feet 2 inches; what is its diameter ? 13. The circumference of a circle is 154 feet 11 inches; find its diameter. Section 5. 14. The two wheels of a coach are 4 feet 6 inches, and 4 feet in diameter; how many times will each turn in a journey of 20 miles ? Section 5-(continued). 15. The circumference of a steam-engine cylinder is 13 feet 1 inch; find its diameter. 16. The circumference of a well is 16 feet 9 inches; what is its diameter. Section 6. 17. A railway-train moves, up an inclined plane of 3 in 396, at the rate of 37 miles an hour; what vertical height will it ascend per minute? 18. A railway-train moves, up an inclined plane of 7 in 1071, at 28 miles per hour; what vertical height will it ascend per hour? 19. An inclined plane rises 18 yards in 3 miles; in how many yards will it rise 1 yard? 20. Two vessels, 7 miles distant, the one sails in a direction of 1 to 34, and the other 1 to 43, with the line joining their original positions; how far will each sail before it crosses the other's direction. PROBLEM 4. Given the chord A B and versed sine C D of an arc of a circle; to find its diameter. EXAMPLES. Section 1. 1. The chord of an arc is 12 feet, and the versed sine is 2 feet; find the diameter. 2. The chord of an arc is 18 feet 6 inches, and the versed sine 5 feet 6 inches; find the diameter. 3. The span of a bridge is 30 feet 6 inches, and its height in the middle, from the springing line, is 8 feet 9 inches; find the radius of the centering. Section 2. 4. The span of the middle arch of Dunkeld-bridge is 90 feet, and rise of arch 30 feet; required the radius of the arch. 5. The bridge at Wearmouth is 240 feet span, and rises 30 feet in the middle; what is the radius of the arch? 6. The span of the Buildwas-bridge, near Colebrook Dale, is 130 feet, and height of the versed sine 17 feet; required the radius of the arch. Section 3. 7. The span of the wooden bridge on the Regnitz, in Germany, is 208 feet, with a rise of 16 9 feet; required the radius of the arch. 8. The span of the bridge over the Don, 7 miles above Aberdeen, is 109 feet, the rise 13 feet; required the radius. 9. The bridge at Walton on the Thames has a span of 130 feet, and a rise of 27 feet; required the radius. Section 4. 10. The bridge at Ettringen, over the Wertach, has a span of 138 feet, and a radius of curvature of 305 feet; find the rise in the middle. 11. The bridge of Augsburg, on the Lech, has a span of 114 feet, and a radius of curvature of 158 feet; find the rise in the middle. 12. The bridge of Freysingen, on the Isar, has a span of 153 feet, and the radius of curvature 246 feet; find the rise of the arch in the middle. PROBLEM 5. Given the chord A B, the versed sine C D of a circle, the distance DE; to find the ordinate E F. Put r = radius, the other quantities as in Problem 4. :. E F = √(r — D E) (r + D E) — (r — v). EXAMPLES. Section 1. 1. The chord and versed sine of an arc of a circle are 14 feet and 6 feet respectively; find the ordinates at 1, 2, and 3 feet from the middle of the chord. Section 2. 2. A railway bridge, single line, is 36 feet span and 9 feet rise : the diameter of the funnel of a locomotive is 1 foot 6 inches, and the level of the rail below the springing of the arch is 16 feet; what is the greatest height of the engine and funnel the bridge will allow ? Section 3. 3. A tunnel for a double line of railway is 22 feet span, and rise 18 feet; the rails are 5 feet below the springing line, and the outer one is 8 feet 11 inch from the centre of the tunnel; find the highest carriage which the tunnel will take. PROBLEM 6. From a given distance A B along the tangent to a circle, parabola, and ellipse; find the deflection B P to the curve. Circle. Rule.-Take the sum and difference of the radius and given distance (A B), multiply them together and extract the square root; then subtract this quantity from the radius, and the result will be the deflection. B Parabola. Rule.-Square the distance A B, and divide this result by four times the distance from the vertex to the focus, and the quotient will be the deflection. Square half the chord of the parabola, and divide the result by the height or versed sine of the curve; the quotient is four times the focal distance. Ellipse. Rule.-Take the sum and difference of the semiaxis minor and the distance (A B), multiply them together and extract the square root of the product, subtract this quantity from the semiaxis minor; multiply the result by the semiaxis major, and divide the product by the semiaxis minor, and the quotient will be the deflection. When the deflection of curves is small, they may be considered portions of a parabola without producing any appreciable error. |