120. On the bank of a river there is a column 200 feet high supporting a statue 30 feet high. The statue to an observer on the opposite bank subtends an equal angle with a man 6 feet high standing at the base of the column: find the breadth of the river. Ans. 10√115 feet.

121. A man walking along a straight road at the rate of 3 miles an hour, sees in front of him, at an elevation of 60°, a balloon which is travelling horizontally in the same direction at the rate of 6 miles an hour; ten minutes after he observes that the elevation is 30°: prove that the height of the balloon above the road is 440√3 yards.

122. An observer in a balloon observes the angle of depression of an object on the ground, due south, to be 35° 30'. The balloon drifts due east, at the same elevation, for 2 miles, when the angle of depression of the same object is observed to be 23° 14': find the height of the balloon. Ans. 1.34394 miles.

123. A column, on a pedestal 20 feet high, subtends an angle 45° to a person on the ground; on approaching 20 feet, it again subtends an angle 45°: show that the height of the column is 100 feet.

124. A tower 51 feet high has a mark 25 feet from the ground: find at what distance the two parts subtend equal angles to an eye 5 feet from the ground. Ans. 160 feet.

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125. From the extremities of a sea-wall, 300 feet long, the bearings of a boat at sea were observed to be N. 23° 30' E., and N. 35° 15′ W.: find the distance of the boat from the sea-wall. Ans. 262.82 feet.

126. ABC is a triangle on a horizontal plane, on which stands a tower CD, whose elevation at A is 50° 3' 2"; AB is 100.62 feet, and BC and AC make with AB angles 40° 35' 17" and 9° 59' 50" respectively: find CD. Ans. 101.166 feet.

D

127. The angle of elevation of a tower at a distance of 20 yards from its foot is three times as great as the angle

of elevation 100 yards from the same point: show that the height of the tower is

300

feet. √7

128. A man standing at a point A, due south of a tower built on a horizontal plain, observes the altitude of the tower to be 60°. 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.

=

30

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129. The angle of elevation of a balloon, which is ascend- & ing 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.

130. The angle of elevation of the top of a steeple at a place due south of it is 45°, and at another place due west of the former station and distant 100 feet from it the elevation is 15°: show that the height of the steeple is 50(31 — 3-1) feet.

131. A tower stands at the foot of an inclined plane whose inclination to the horizon is 9°; 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 feet.

132. The altitude of a certain rock is observed to be 47°, 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 feet.

133. From a window it is observed that the angle of elevation of the top of a house on the opposite side of the street is 29°, and the angle of depression of the bottom of the house is 56°: find the height of the house, supposing the breadth of the street to be 80 feet. Ans. 162.95 feet.

134. 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 60°: prove that the distance between A and B is 9.165 miles.

135. P and Q are two inaccessible objects; a straight line AB, in the same plane as P and Q, is measured, and found to be 280 yards; the angle PAB is 95°, the angle QAB is 471°, the angle QBA is 110°, and the angle PBA is 52° 20' : find the length of PQ. Ans. 509.77 yards.

136. Two hills each 264 feet high are just visible from each other over the sea: how far are they apart? (Take the radius of the earth = 4000 miles.) Ans. 40 miles.

137. A ship sailing out of harbor is watched by an observer from the shore; and at the instant she disappears below the horizon he ascends to a height of 20 feet, and thus keeps her in sight 40 minutes longer: find the rate at which the ship is sailing, assuming the earth's radius to be 4000 miles, and neglecting the height of the observer.

Ans. 40√330 feet per minute.

138. From the top of the mast of a ship 64 feet above the level of the sea the light of a distant lighthouse is just seen in the horizon; and after the ship has sailed directly towards the light for 30 minutes it is seen from the deck of the ship, which is 16 feet at which the ship is sailing.

above the sea: find the rate (Take radius = 4000 miles.)

Ans. 8 miles per hour.

139. A, B, C, are three objects at known distances apart; namely, AB = 1056 yards, AC = 924 yards, BC = 1716 yards. An observer places himself at a station P from which C appears directly in front of A, and observes the angle CPB to be 14° 24': find the distance CP.

Ans. 2109.824 yards.

140. A, B, C, are three objects such that AB=320 yards, AC 600 yards, and BC=435 yards. From a station P it is observed that APB = 15°, and BPC = 30°: find the distances of P from A, B, and C; the point B being nearest to P, and the angle APC being the sum of the angles APB and BPC. Ans. PA=777, PB = 502, PC = 790.

CHAPTER VIII.

CONSTRUCTION OF LOGARITHMIC AND TRIGONOMETRIC TABLES.

128. Logarithmic and Trigonometric Tables. In Chapters IV., V., and VII., it was shown how to use logarithmic and trigonometric tables; it will now be shown how to calculate such tables. Although the trigonometric functions are seldom capable of being expressed exactly, yet they can be found approximately for any angle; and the calculations may be carried to any assigned degree of accuracy. We shall first show how to calculate logarithmic tables, and shall repeat here substantially Arts. 208, 209, 210, from the College Algebra.

129. Exponential Series. To expand e* in a series of ascending powers of x.