Example. The top of a ship's mast is 663 ft. above the sea-level, and from it the lamp of a lighthouse can just be seen. After the ship has sailed directly towards the lighthouse for half-an-hour the lamp can be seen from the deck, which is 24 ft. above the sea. the rate at which the ship is sailing. Let L denote the lamp, D and E the two positions of the ship, B the top of the mast, C the point on the deck from which the lamp is seen; then LCB is a tangent to the earth's surface at A. [In problems like this some of the lines must necessarily be greatly out of proportion.] Let AB and AC be expressed in miles; then since DB=66 feet and EC 24 feet, we have by the rule = 2×663=100; Find But the angles subtended by AB and AC at O the centre of the earth are very small; .. arc AD=AB, and arc AC=AE. .. arc DEAD-AE=AB-AC-4 miles. [Art. 268.] Thus the ship sails 4 miles in half-an-hour, or 8 miles per hour. 275. Let be the number of radians in the dip of the horizon; then with the figure of Art. 273, we have neglect the terms on the right after the first. Thus or Let N be the number of degrees in 6 radians; then Now r=63 nearly; hence we have approximately a formula connecting the dip of the horizon in degrees and the height of the place of observation in miles. 1. Find the greatest distance at which the lamp of a lighthouse can be seen, the light being 96 feet above the sealevel. 2. If the lamp of a lighthouse begins to be seen at a distance of 15 miles, find its height above the sea-level. 3. The tops of the masts of two ships are 32 ft. 8 in. and 42 ft. 8 in. above the sea-level: find the greatest distance at which one mast can be seen from the other. 4. Find the height of a ship's mast which is just visible at a distance of 20 miles from a point on the mast of another ship which is 54 ft. above the sea-level. 5. From the mast of a ship 73 ft. 6 in. high the lamp of a lighthouse is just visible at a distance of 28 miles: find the height of the lamp. 6. Find the sexagesimal measure of the dip of the horizon from a hill 2640 feet high. 7. Along a straight coast there are lighthouses at intervals of 24 miles: find at what height the lamp must be placed so that the light of one at least may be visible at a distance of 3 miles from any point of the coast. 8. From the top of a mountain the dip of the horizon is 1·81°: find its height in feet.、 9. The distance of the horizon as seen from the top of a hill is 30.25 miles: find the height of the hill and the dip of the horizon. 10. If x miles be the distance of the visible horizon and N degrees the dip, shew that 14. Two sides of a triangle are 31 and 32, and they include a right angle: find the other angles. 15. A person walks directly towards a distant object P, and observes that at the three points A, B, C, the elevations of P are a, 2a, 3a respectively: shew that AB=3BC nearly. CHAPTER XXII. GEOMETRICAL PROOFS. 276. To find the expansion of tan (A+B) geometrically. Let ▲ LOM=A, and ▲ MON=B; then ▲ LON=A+B. In ON take any point P, and draw PQ and PR perpendicular to OL and OM respectively. Also draw RS and RT perpendicular to OL and PQ respectively. also the triangles ROS and TPR are similar, and therefore In like manner, with the help of the figure on page 95, we may obtain the expansion of tan (A-B) geometrically. 277. To prove geometrically the formula for transformation of sums into products. Let EOF be denoted by A, and LEOG by B. With centre O and any radius describe an arc of a circle meeting OG in H and OF in K. Bisect KOH by OL; then OL bisects HK at right angles. Draw KP, HQ, LR perpendicular to OE, and through L draw MLN parallel to OE meeting KP in M and QH in N. It is easy to prove that the triangles MKL and NHL are equal in all respects, so that KM=NH, ML=LN, PR=RQ. |