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another, and found to be o and respectively. Show that the height of the second tower

tan o

tano 10. From the top of a tower the angles of depression of two objects in a direct line, and whose distance from each other is a, are a, ß respectively. Show that the height of the tower

tan •

a

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2 a

11. A person, having walked a distance a from one corner along a side of an oblong, observes that the side immediately behind him subtends an angle a, and the side in front an angle B.

Show that the dimensions of the oblong are

a tan a (1 + tan a cot B). 12. Three points A, B, C form a triangle whose sides are a, b, c respectively, and a person standing at a point S, such that SA is at right angles to BC, observes that the side AC subtends an angle 0. Show that the distance of S from B

1

Na> + c2 – 62)2 + (a? + b2 — co) coť o. 13. A person walks a yards from A to E along AB the side of a triangle ABC, and observes that the angle AEC

= a; he also walks b yards from B to F along BA, and observes L CFB B. Having given AB = c, find BC and AC.

14. A tower is observed from three stations A, B, C, in a straight line not meeting the tower, to subtend angles a, b, y respectively. Show that if AB = a, BC = 6, the height of the tower

ab(a + b)

a cota y — (a + b) cotß + b cot? When are the conditions impossible?

15. From two stations whose distance apart is a, and which are due W. and due S. respectively of one end of a wall, the angles subtended by the wall are each Show that the length of the wall is a sin a.

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a

a.

COS 0

16. The angles of elevation of the top of a tower, whose height is h and standing on a hill, area, ß, when observed from two stations a miles distant, and in a direct line up the hill, Show that if o be the slope of the hill —

a sin a sin B

Ti sin (B – a) 17. The elevation of a tower was observed to be but on walking in the horizontal plane a distance a at right angles to the line joining the first position and the foot of the tower, the elevation was ß. Show that the height of the tower was

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a

cota
ß

cota 18. The angles of depression of two objects in the same horizontal plane, as seen from the top of a tower, are o and respectively, and the angle they subtend is a. Show that if h be the height of the tower, the distance between the objects—

h cosec 0 + cosec 2 cosec 0 cosec o cos a.

ANSWERS,

1.- PAGE 15.

1. 45.23, 290, 2367, 7, &c.
2. .0005, 11•11, 040020, 45, &c.

3. Three thousand four hundred and sixty-seven thousandths; thirty-four, and sixty-seven hundredths; three thousand four hundred and sixty-seven millionths; three, and four hundred and sixty-seven thousandths.

4. 35.90846, 29130.19391, 60·0239. 5. 7237, 3.32091. 6. 69.5289, 5.06679, •41481. 7. •026, 7708·71. 8. 09, 24:356706, 003627, 289, .0096, .00016384. 9. 200, .00125, 4000, .2295, •006, •5002, &c. 10. 170000. 11. 4.97 &c., 1. 12. •2.

7

10

17

999

64

6

II.-PAGE 23. 1. *, 4:3, 1007, 607, 1010, 729. 2. 114, 42, 43, 4, 19, 28. 3. 6, 7, 12, 11, 9, 14; 4, , 6, 42, , 4*. 4. 77, 13, 73, 163, 713, 314, &c. 5. 52. 6. 15, , , 33, 174, 114 ; 17, &c. 7. 3, 31, 34; }}, , ! ; ;', $. 8. 41, 6, 1, 1, 6. 9., 23,9}, If, 1, 3. 10. 111.

11. 3. 12. 28.

III. - PAGE 28.

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1. 4 x 11, 2°, 2 x 3 x 57, 22 x 32 x 7, 33 x 47, 2 x 3 x 7 x 11.

2. 2 x 7 x 32 x 11, 23 x 5 x 43, 29 x 32, 52 x 7 x 11, 22 x 3 x 11', 24 x 34 x 11.

5. , , , tt, , $; js, f., 1., , }, }); }, , s 1%, 1, abbi.

6. 19, 7, 13, 6, 41, 729, 14, 11, 39. 7. , 7, it is, 1%, $; \, 4, 1}, }, {}, } 9. 2 days out of every 7. 10. 2: 7. 12. 15, 10, 6.

IV. - PAGE 31.
1. 24, 1260, 2520, 1260, 5460, 33300.

2. 46, 48, , ; it, ii, ii, 14 ; , , fit, i} }; 10, 105, 105; 28, 19, 20, ; 387, 38, 3. 12. 4. 1, 1, 1, 18. 5. 10654. 7. 13.

10. 1, }, ".

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V.-PAGE 34.

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VI.-PAGE 36.
2. 5100

3. 5.
5. 327.

6. 4. 8. 244, 15701 10. £8. 121 s. 12. 2015

9. 77. 11. £352. 6s. 8d.

VII.- PAGE 41.

1. 2.203125. 3. 3•703059239. 6. 1. 9. •367879. 11. 3:141592.

2. zo, zbo, } }, }, * .
4. 5.098809263: 5: 156, :
17. 2:718281.

8. •321750.
10. 1.015873.
12. •857142.

VIII.-PAGE 43.

1. 13s. 4d., 7d., 2s. 8d., £1. 5s. 84d. 2. £4. 7s. 6d., 6., ls. 11d., 3s. 9d. 3. £5. 178. 4d., £8. 15s., £25. 3s. 1114d. 4. 17 cwt. 16 lbs., 164 lbs., 63 lbs., 21 lbs. 5. 3 m. 2 f. 62. yds., 293} yds., 4 po. 14 yd., 823* yds. 6. 144 days, 32 days, 4 hrs. 54' 47'. 17. 38 lbs. 7 oz. 2 dwt. 159 grs., 12 dwt. 670 grs. 8. - 245 ac. 1 r. 27 po. 12 P's yds. 9. 40° 3' 28", 35°. 10. 6153 grains. 11. 711 ; 4cwt. 1 qr. 143 lbs. 12. 1 day 3 hrs. 8' 2413187":

IX.

PAGE 45.

2 2 9 2688.

1. 5, 16 2. j'a, 1 .

3. alio, 23 4. 3 lbs. 7 oz. 15 dwts., 81245 lbs. 5. 46,

6. 114, 11: 7. , 3Y

8. 1185, 11:30: 9. 1828361

10. %.
12. to day of 24 hrs.

11. 68

X-PAGE 47.

1. 78. 6d., 19s. 7 d., 16s. 3}d. 2. £1. 5s., 2s. 9d., £1. 6s. 3d.

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