(12) A and B are two positions on opposite sides of a mountain; C is a point visible from A and B; AC and BC are 10 miles and 8 miles respectively, and the angle BCA is 60o. Prove that the distance between A and B is 9·165 miles.

(13) In the last question, if the angle of elevation of Cat A is 8o, and at B is 2o 48′ 24′′: show that the height of A above B is one mile very nearly.

sin 801391731 sin 2° 48′ 24′′0489664.

(14) Show that the angles which a tunnel going through the mountain from A to B, in Questions (12) and (13), would make (i) with the horizon, (ii) with the line joining A and C, are respectively 6° 16′ and 49° 6′ 24′′.

sin 6o 16'1091; tan 10° 53′ 36′′='192450.

(15) A and B are consecutive milestones on a straight road; C is the top of a distant mountain. At A the angle CAB is observed to be 38° 19′; at B the angle CBA is observed to be 132o 42', and the angle of elevation of C at B is 10° 15'. Show that the top of the mountain is 1243-5 yards higher than B.

I sin 38° 19' 9.7923968

=

L cosec 8° 59′10.8064659

L sin 100 15′=9-2502822.

log 1760-3.2455127 log 1243.5-3.09465

(16) A base line AB, 1000 feet long is measured along the straight bank of a river; C is an object on the opposite bank; the angles BAC, and CBA are observed to be 65° 37′ and 53° 4′ respectively. Prove that the perpendicular breadth of the river at C is 829-87 feet; having given

L sin 65° 37' =
=

9.9594248,

I cosec 61° 19' = 10.0568589,

L sin 53° 4'=9.9027289

log 8.2987-91901.

* MISCELLANEOUS EXAMPLES. LXXII,

(1) A man walking along a straight road at the rate of three miles an hour sees, in front of him at an elevation of 60o, a balloon which is travelling horizontally in the same direction at the rate of six miles an hour; ten minutes after he observes that the elevation is 30o. Prove that the height of the balloon above the road is 440/3 yards.

(2) A person standing at a point A, due south of a tower built on a horizontal plain, observes the altitude of the tower to be 60o. He then walks to a point B due west from A and observes the altitude to be 45°, and then at the point C in AB produced he observes the altitude to be 30°. Prove that AB=BC.

(3) The angle of elevation of a balloon, which is ascending uniformly and vertically, when it is one mile high is observed to be 35° 20′; 20 minutes later the elevation is observed to be 55° 40'. How fast is the balloon moving?

Ans. 3 (sin 20° 20′) (sec 55° 40′) (cosec 35° 20′) miles per hour.

(4) The angular elevation of a tower at a place A due south of it is 30°; and at a place B due west of A, and at a distance a from it, the elevation is 18o; show that the height of the tower is

α

√2(1+√5)

(5) The angular elevation of the top of a steeple at a place due south of it is 45o, and at another place due west of the former station and distant a feet from it the elevation is 15o; show that

α

the height of the steeple is (3-3-1) feet.

(6) A tower stands at the foot of an inclined plane whose inclination to the horizon is 9o; a line is measured up the incline from the foot of the tower of 100 feet in length. At the upper extremity of this line the tower subtends an angle of 54°. Find the height of the tower. Ans. 114.4 ft.

(7) The altitude of a certain rock is observed to be 47o, and after walking 1000 feet towards the rock, up a slope inclined at an angle of 32° to the horizon, the observer finds that the altitude is 77°. Prove that the vertical height of the rock above the first point of observation is 1034 ft. Sin 47°='73135.

(8) At the top of a chimney 150 feet high standing at one corner of a triangular yard, the angle subtended by the adjacent sides of the yard are 30o and 45° respectively; while that subtended by the opposite side is 30o. Show that the lengths of the sides are 150 ft. 86.6 ft. and 106 ft. respectively.

(9) A flag staff h feet high stands on the top of a tower. From a point in the plain on which the tower stands the angles of elevation of the top and bottom of the flagstaff are observed to be a and ẞ respectively. Prove that the height of the tower h tan B h sin ẞ.cos a tan a-tan B sin (a-B)

is

feet, i.e.

feet.

(10) From the top of a cliff h feet high the angles of depression of two ships at sea in a line with the foot of the cliff are a and B respectively. Show that the distance between the ships is h (cot - cot a) feet.

(11) The angular elevation of a tower at a place due south of it is a, and at another place due west of the first and distant d from it, the elevation is ß. ́ Prove that the height of the tower is

(12) A man stands on the top of the wall of height h, and observes the angular elevation (a) of the top of a telegraph post; he then descends from the wall, and finds that the angular elevation is now ẞ; prove that the height of the post exceeds the height of sin B. cosa sin (ß − a) *

the man by h

(13) Find the height of a cloud by observing (i) its elevation (a) and (ii) the depression (B) of its reflexion in a lake h feet below the point of observation. Ans. Height above the lake h sin (a + B) cosec (B − a) feet.

(14) If the angular elevation of the summits of two spires (which appear in a straight line) is a, and the angular depressions of their reflexions in a lake, h feet below the point of observation, are B and 1, then the horizontal distance between the spires is 2h cos2 a sin (B-y). cosec (ẞ—a). cosec (y—a) feet.

(15) A statue AB on the top of a pillar BC situated on level ground is found to subtend the greatest angle (a) at the eye of an observer E when he is distant c feet from the pillar. Prove that the height of the statue is (2 c tan a) feet; and, if b be the height of the observer's eye from the ground find the height of the pillar. (See Note in the Answers.)

(16) A man walking along a straight road observes that the greatest angle which two objects subtend at his eye is a. From the place where this happens he walks a yards; the objects then appear in a straight line making a right angle with the road. Prove that the distance between the objects is (2 a tan a) yards.

(17) A man standing on a horizontal plain at a distance h feet from a tower, observes that a flagstaff on the tower subtends an angle a at his eye, and that on walking 2 k feet towards the tower the flagstaff subtends again the same angle, Prove that the height of the flagstaff is 2 (h-k) tan a feet.

(18) A man walking along a straight road observes that the greatest angle which two objects subtend at his eye is a; from the place where this happens he walks a yards, and the objects then appear in a straight line making an angle ẞ with the road. 2a sin a. sin B

Prove that the distance between the objects is

cos a + cos B

yds.

CHAPTER XIX.

ON TRIANGLES AND CIRCLES.

273. To find the Area of a Triangle.

The area of the triangle ABC is denoted by A.

Through A draw HK parallel to BC, and through ABC

draw lines AD, BK, CH perpendicular to BC.

The area of the triangle ABC is half that of the rectangular parallelogram BCIIK

[Euc. 1. 41].