Mathematical Exercises ...

9. A U-shaped tube of uniform bore, of which the curved part is a semicircle 12 inches in length, is placed in an upright position, and contains as much mercury as fills 15 in. of its length, and as much water above the mercury on one side as fills 41 in. more. If the difference of level between the two mercurial surfaces is 3 in. nearly, what is the s.g. of mercury, and what is approximately the height of a column of water, which will produce a pressure equal to that at the bottom of the tube?

10. Prove that, when a body floats in a fluid, the weight of the fluid it displaces is equal to the weight of the body.

11. Show that the fluid pressure on any surface immersed in a uniform fluid at rest under the action of gravity, will be equal to that on a plane surface of equal area immersed in the same fluid, in a horizontal position, and at the same depth as the centre of gravity of the proposed surface.

12. A hollow cube, with four of its sides vertical, is filled with water. If one of the vertical sides were to become entirely detached from the base and the two adjoining sides, find the magnitude, direction, and point of application of the single force which would preserve the unattached side in its position.

(4) 1. £180892 15s. 03d.; 2. £862287 5s. 7 d.

(5) 1. £12204 14s. 2d.; 2. £78

(6) £414 13s. old.

(8) 2 cwt. 3 qrs. 14 lbs.

(10) 1. £1060 6s. 74d.; 2. £419 3s. 41d.

(11) £293 88. 5'4d.

(3) £2282 11s. 6d.

4s. 112d.

(7) 1.673328; 2. 192669.

(9) 427 oz.

(12) 4.

(2) 2. £558308 19s. 03d.

(4) 1.3 yds. 20 ft. 1365 in.; 2. 46 yds. 5 ft. 127 in.

(5) 1. 16; 2. 26.

(7) 32.

(8) 1. £343 12s. 4d.; 2. £1281 19 0

(9) 1. 015625; 2. 4255952380. (10) £46553. (11) £94 3s. 4d.

(12) £750.

(3) £87 13s. 8d.

(6) 16.

(1) 1. 78256; 2. 29 mi. 1 fur. 63 yds. 10 in.; 3. 2 yds. 26 ft. 57 in. (3) 1. 17; 2. £3 12s. 2d.

(2) £36 10s. 21d.

(4) 078125; 2142857; 23,600 16000

(5) 148809523...; 3.

(8) 45 days. (11) 140 yds.

(1) 2. 36430031911. (2) 85001997; 885768804.

(4) 2. £351154 5s. 10d. (5) 60.

(7) £74 7s. 6d.

(9) £7 15s. 2 d.

(6) 112640.

(8) 1. £132 2s. 3121d.; 2. £625 19s. Ogd.

(10) 3113.

229

(11) 1. £4 28. 133d.; 2. 970 57005897.

XII.

(12) 40000.

(1) 1. £1199679 6s. 3d.; 2. 45 mi. 7 f. 25 p. 2 yds.

(10) £500.

(11) £35049 12s.; £81782 8s.; £105148 16s.; £128515 48.

(12) 2256 dol. 96 c.

(1) 2.125550; Remainder =79. (2) £10656 5s.

(3) 23.

(4) 2 ro. 27 po.

(6) 1.1; 2.1.

(8) £513.

(10) 115 ft. (12) 2907.

13 14
(5) 1. 1211; 2. 2; 3.72; 4. > by

79
82

77 88 1232

(7) 1. £1156 Os. 7d.; 2. £30 19s. 4d.
(9) 282 cwt. 2 qrs.

(11) £40 10s.; £46 10s.; £67 10s.

(2) 51964; 1643.

(3) 75 tons 5 cwt. 1 qr. 25 lbs.

(5) 2331; 19 m. 3 fur. 193 yds. 1 ft. 7 in.

(7) 1. ; 2. 318; 3.5; 4. 28.

(8) 30129; 14s. 9 d.

(10) £6 8s. 4d. (11) 27 days.

(1) 154096 oz.; 396 ells.

(6) 1. 1097; 2. 80; 3. 3128.

(8) 1. £1246 16s. 6d.; 2. £15 5s. 4 d. (10) 78. 6d.

(7) 10s. 1d. =

97 193'

(9) £4 7s. 6d.;

7d. per lb.

(11) 27 days.

XIX.

(12) 6 ac. 293 yds. 3 ft.

(1) 1. 757 tons 3 qrs.; 2. £884818 6s. 7d.; 3. 9 m. 4 f. 39 p. 238 ft. (2) 1.31622400; 2. £2 5s. 7d.; £4 11s. 2d.; 3. 562243. 2. £47 16s. 3d.; 3. £21 11s. 3d.; 8d.

(3) 1. £25 11s. 4d.;