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EXERCISES IN CHAPTER VII, PROBLEM V.

1. Hyp. = 12%. Perp. = 81. The angles are required. 2. Base=0.8. Perp. = 1·2. What are the angles ?

3. The base of an isosceles triangle is 5, and each of its equal sides double the base. What is the vertical angle?

EXERCISES in CHAPTER VII, PROBLEM VI.

1. The legs being 2 and 3, what are the acute angles ?

2. Base=3 yds. 1 ft. 6 in. Perp. = 5 yds. 9 in. What are the acute angles ?

3. What is the angle of inclination of a gradient rising at the rate of 1 in 28 ?

4. The legs being .0096 and ·0192, what are the acute angles ?

EXERCISES IN CHAPTER VII, PROBLEM VII. ,

1. Hyp. = 8 ft. 54 in. Base = 6 ft. 44 in. Perp. required.

2. Hyp. = 8125. Perp. = 3125. Base required. 3. Hyp. = 0.13167. Base = .02660. Perp. required.

EXERCISES IN CHAPTER VII, PROBLEM VIII.

1. Base = 12.6. Perp. = 1.33. Hyp. required. 2. Base = 26. Perp. = 128.7. Hyp. required. 3. Legs •0784 and .0377. Hyp. required.

PROMISCUOUS EXERCISES IN RIGHT-ANGLED TRIANGLES.

1. Hyp.= 101. Perp. = 99. Base required.

2. One of the equal sides of an isosceles triangle is 19:49, and the angle at the vertex 45°. What is the base ?

3. Compute, by Trigonometry, the side of an equilateral triangle whose altitude is 100.

4. The three sides of a triangle are 193, 193, and 190. Taking the shortest side as the base, what is the altitude, and what are the angles at the base ?

5. The largest of the pyramids of Egypt, being more correctly a frustum of a pyramid, is 480 feet high, and stands on a base 745 feet square, its summit being 32 feet square. At what angle are the sides inclined to the horizon?

6. One of the acute angles of a right-angled triangle is four times the other. The shorter leg is 10. What is the longer leg, and the hypothenuse ?

EXERCISES IN CHAPTER VIII, PROBLEM I.

1. Given, the angles at the base 33 and 591 Required the angle at the vertex.

2. ZA= 84° 49'. ZB=63° 28'. ZC is required.

3. ZB=49° 10' 53". ZC=72° 57' 15". ZA is required.

4. The sum of the angles at the base of a triangle is 114° 48'. What is the angle at the vertex ?

5. The angle at the vertex being 771°, what is the sum of the angles at the base ?

EXERCISES IN CHAPTER VIII, PROBLEM II.

1. Given, ZA=49° 54', ZB=72° 28', side a=769. Required side b.

2. The three angles of a triangle are respectively 54°, 69°, and 57°. The shortest side is 8.58 chains. What are the other two sides?

3. Two angles of a triangle are severally 39° 10' and 104° 48'. The side opposite the former is 48 yds. 2 ft. What is the side opposite the latter?

4. What is the third side of the last-mentioned triangle?

EXERCISES IN CHAPTER VIII, PROBLEM III.

1. Side a=4.94. Side b=5.09. ZB=118°. ZA is required.

2. Sides a and b= 5 and 6. ZA=541°. ZB is required, which is acute.

3. Sides a and b=0.3 and 0'4. ZA= 38° 43'. ZB is required, which is obtuse. 4. Side a=

= 548.3. Side b=499.4. ZA=76° 23'. What is ZB?

5. Side a=10. Side b=11. ZA= 57° 4'. What is LB?

EXERCISES IN CHAPTER VIII, PROBLEM IV.

Compute the third side in each of the five triangles described in the Exercises under the preceding Problem.

EXERCISES IN CHAPTER VIII, PROBLEM V.

1. Sides, 4 and 5. Included angle, 66°. What are the other angles ?

2. Sides, 8f and 9. Vertical angle, 182o. The other angles are required.

3. Sides, 23 and 21. Included angle, 96° 20'. Compute the other angles.

EXERCISE IN CHAPTER VIII, PROBLEM VI.

Calculate the third side of each of the three triangles described in the preceding Exercises, under Problem v.

EXERCISES IN CHAPTER VIII, PROBLEM VII.

1. The three sides being 1636, 1524, and 1540, what are the three angles ?

2. Sides, 11•46, 9•68, and 8:674. What are the angles ?

EXERCISES IN CHAPTER IX.

NOTE. Instead of giving a new set of questions in this place, it has been thought better merely to give repetitions of those in Chapter 1x, with the figures altered. Thus, of the following questions, the first is understood to be the same as Exercise 1 of that Chapter, substituting 12 degrees for 15, and 36 feet for 50.

1. 2. 3. 4. 5. 6.

.12 degrees...... ..36 feet.........? 1 in 100;

? 14 degree,

? 18 miles West...... 20 miles South...... ? 53......... 27........201... ...... ? ..NNW, 12 miles ..

........

......

.......

7.

E by N, 86 miles ........ ? 8. ... ...... between S and W, 200 miles....... 841 miles :

. ? 10. ...315............10° 38':

? 11. 87 feet, ...... 19° 59'.

? 12.

12 feet......25 feet......30 inches lower......? 13. .... Latitude 51° 25'........ .75 feet ....... ?

16. ....... 24 feet square.......36 yards...... 33° 39'...... 161 feet.........?

17. ....... 28° 38'.......5 feet .... .12° 12' .......496 feet. .........? 18. ...101 feet......34° 45'...... 20 feet above the base

? 19. .... .8° 14' ;.........78° 22' ...........168 feet,... ? 20. ......N 22° E, ......N 28 W ......300......200...... ? 21.

.ESE ;........E by N........SW by S........500 yards

? 24. .........108......... angular depression of the top...... 2° 49', and of the base 26° 52'

?
26. .........1 mile..........:-
ZABC=54° 16'.

ZDBC= 64° 20'.
ZBAC=73° 28'.

Z CDE= 58° 59'.
ZBCD= 61° 33'.

ZDCE = 56° 11'.
DE is required in feet.

27. .... 15° 31'.........300 feet......... 21° 54'.........

29. ......6° 3' and 8° 30'......8 chains......0° 44'.... the more distant station being the lower.

30. .......8 chains.......3° 50' and 7° 29' ....... 26 feet higher...... 12 feet higher...... ?

31. ......5 feet......1 chain......38° 10'......1 chain...... 21° 42'. .?

32. ......... The base EF of 100 feet.........86° 18' and 70° 47'; and the angle PER, 15° 42' ...... ?

34. ....... a base of 30 chains....... angle of elevation at E = 2° 53'. Horizontal angle at E= 86o. Horizontal angle at F= 91° 13'......... 36. .............12.88 chains ;............91° 36' and 84° 2'.

.2° 12'. 41. ...........36 miles per hour ..........5 minutes after ......... 24° 42'. What was the height of the cloud in feet?

43. ......... 71°.........45°.........657 yards. What was the height of the cloud in feet? 44.

.2 miles.......... ? 51. .PQ = 2, QR = 3, PR = 4.........45° ........?

81

TABLES.

Table I.-Logarithms of Numbers from 1 to 250, with their Indices.

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1 0.000000 51 1707570| 101 2.004321 151 2:178977| 201 2.303196 20:301030 52 1.716003 102 2:008600 152 2:181844202 2.305351 30.477121 53 1:724276 103 2.012837 1532-184691 203 2.307496 40.602060 54 1.732394 104 2.017033| 154 2.187521 204 2.309630 5 0.698970 55 10740363 105 2.021189 155 2.190332 205 2:311754 60:778151 56 1.748188|| 106 2.025306 156 2.193125 206 2:313867 70 845098 57 10755875 107 2.029384 157 2:195900 207 2:315970

8 0.903090 58 1.763428 108 2.033424 1582-198657 208 2:318063 90.954243 59 1.770852 109 2:037426 159 2.201397 209 2:320146 10|1.000000 60 1.778151 110 2:041393 1602.204120 210 2:322219 11 1.041393 61 1.7853301112-045323 161 2.206826 211 2:324282 121.079181 62 1.792392 112 2:049218 162 2.209515| 212 2:326336 131.113943 63 1.799341 113 2:053078| 163 2.212188 213 2:328380 14 1.146128 64 1.806180|| 114 2:056905 164 2.214844 214 2:330414 15 1.176091 65 1.812913 1152.060698 1652-217484 215 2:332438 16 1.204120 66 1.819544 116 2064458 166 2.220108 216 2:334454 17 1.230449 67 1.826075 117 2.068186 167 2.222716 217 2.336460 18 1.255273 68 1.832509 118 2.071882 168 2.225309 218 2:338456 19 1.278754 69 1.838849 119 2.075547| 169 2.227887| 219 2 340444 20 1301030 70 1.845098 120 2.079181 1702-230449| 220 2:342423 21 (1:322219 71 1.851258 121 2.082785 171 2.232996 221 2:344392 22 1.342423 72 1.857332122 2.086360 172 2.235528|| 222 2.346353 23 1.361728 73 1.863323 123 2:089905) 173 2.238046 223 2:348305 24 1.380211 74 1.869232 124 2:093422 174 2.240549 224 2:350248 25 1.397940 75 1.875061 125 2.0969101 1752-243038 225 2:352183 26 1.414973 76 1.880814 126 2.100371 176 2.245513 226 2:354108 27 1.431364 77 1.886491 127 2.103804| 177 2.247973 227 2:356026 28 1.447158 78 1.8920951282-107210 178 2.250420 228 2:357935 29 1.462398 79 1.897627 129 2 110590|| 179 2252853| 229 2:359835 30 1.477121 80 1.903090 130 2:113943 180 2.255273 230 2:361728 31 1.491362 81 1.908485 131 2.117271 181 2.257679|| 231 2:363612 32 1:505150 82 1.913814 132 2.120574 182 2.260071 232 2:365488 33 1.518514 831.919078) 1332.123852 183 2.262451 233 2:367356 34 1.531479 84 1.924279 134 2:127105 1842.264818 234 2:369216 35 1.544068 85 1.929419 135 2.130334 185 2.267172 235 2:371068 36 1.556303|| 86 1 934498 | 136 2.133539 186 2.269513|| 236 2:372912 37 1.568202 87 1.939519 137 2:136721 187 2:271842) 237 2:374748 38 1:579784 88 1.944483| 1382.139879| 188 2.274158 238 2:376577 39 1.591065 89 1.949390 139 2.143015 189 2.276462| 239 2.378398 40 1.602060 90 1.954243 140 2:146128| 190 2.278754 240 2:380211 41 1.612784| 91 1.959041 141 | 2:149219 | 191 2.281033|| 241 2:382017 42 1.623249 92 1.963788 142 2:152288) 1922.283301 242 2.383815 43 1.633468 93 1.968483 143 2:155336 193 2.285557|| 243 2-385606 44 1.643453 94 1.973128 144 2.158362 1942.287802 244 2:387390 45 1.653213 95 1.977724 145 2:161368| 195 2.290035 245 2.389166 46 1.662758 96" 1.982271 146 2.164353| 196 2.292256|| 246 2:390935 47 1.672098 97 1.986772 1472 167317| 197 2.294466|| 247 2:392697 48 1.681241 98 1.9912261482-170262 1982 296665| 248 2:394452 49 1.690196 99 1.995635 149 2:173186| 199 2.298853|| 249 2:396199 50 1.698970 100 2.000000 150 | 2:176091 200 2301030|| 250 | 2.397940

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