5. What is the area of the heart CABDFEC, the axis CD 10 inches. (See article on spirals, page 140.)

Ans. 104.7193 inches.

D

6. There are two pulleys AHD, BFE, the diameters AD, and BE are each = 20 inches, L and the distance Cc is 4 feet; the pulley BFE is put in motion around its axis by a belt ABFE-7 DH, passing round AHD. Now if the pulley ILK on the same axis with AHD, is 40 inches in diameter, what must be the diameter hi of a corresponding pulley hfi, around which the belt IlfeKL may pass, so as to be of the same length and tension as that of the belt ABFEDH, and what will be the ratio of the angular velocity of the two pulleys.

K

Draw Ca, ob perpendicular to al, and from c draw cn perpendicular to Ca; cl and cn will be parallel to al, and hence will be al: with the radius Cn describe a circle nrqst, and the tangent nc will be = the tangent al, = √(Cc'— (IC — hc)3). The arc rn is = arc la arc hl.

-

It is required from these data to find the diameter ih. Scholium. In the solution of this problem, it will be necessary to express the arc rn in terms of its functions; the mode of conducting the solution will be left for the student.

7. Required an expression for the superficies, and also for the solidity of a tetrædron ABCD, in terms of its linear edge, AB=A. Ans. A'√3 = the surface. A'√2 the solidity.

=

с

B

8. Required expressions for the surface and solidity of a regular hexædron or cube in terms of its edge. Ans. 6A', its surface. its solidity.

A',

9. How will you express the surface and solidity of an Octædron in terms of Ans. 2A'√3

its edge?

= the surface. A'✓2 the solidity.

10. Let the surface and solidity of a dodecædron be expressed in terms of its edge.

Ans. 15A/(1+5)= the surface.

11. Express in terms of its edge, the surface and solidity of an Icosaedron..

12. The diameter of the Winchester bushel was 18 in ches, and its depth 8 inches: what must the diameter of bushel be when its depth is 7 inches? Ans. 19.1067.

13. Of what diameter must the bore of a cannon be, which is cast for a ball of 24 lbs. weight, so that the diameter of the bore may be 1-10 of an inch more than that of the ball, and supposing a 9 lb. ball to measure 4 inches in diameter ? Ans. 5.646 inches. 14. Suppose the ball on the top of St. Paul's church is 6 feet in diameter, what did the gilding of it cost, at 34d. per square inch? Ans. £237, 10s. ld. 15. What will the gilding of a right rectangular revoloid, whose axis is 4 feet, come to at 5 cents per square inch? Ans. $720.

16. A silver cup, in form of the frustum of a cone, whose top diameter is 3 inches, its bottom diameter 4, and its altitude 6 inches, being filled with beer, a person drank out of it till he could see the middle of the bottom; it is required to find how much he drank?

Ans. 42.899844 cubic inches = .152127 ale gallons, or 1 gill and nearly, the quantity required.

17. Two persons would divide between them, by a plane perpendicular to the base, a hay rick, in the form of a paraboloid, whose altitude is 40, and the diameter of its base 30 feet; it is required to find the difference between the solidities of the parts, supposing the altitude of the section to be 28 feet. Ans. 11265.75803, the difference required.

18. In the construction of a railroad, having contracted for the sum of $1000 to excavate a certain section, the area of whose conjugate sections in 11 different places, taken at equal distances of 3 rods each, including the ends, are as follows, viz: the first, 150 square feet, the second, 160. the third, 165, the fourth, 172, the fifth, 190, the sixth, 210, the seventh, 224, the eighth, 240, the ninth, 202, the tenth, 108, and the eleventh 0. After having disposed of the materials to be excavated at 10 cents per cubic yard, to be delivered on an adjoining section, I afterward received an offer to have the whole labor of excavation and delivery performed for 30 cents per yard; shall I gain or lose by my contract if I accept of the offer, and how much? Ans. I shall gain $355,64. 19. To determine the weight of a hollow spherical iron shell, 5 inches in diameter, the thickness of the metal being one inch? Ans. 11.79lb. 20. It is proposed to determine the proportional quantities of matter in the earth and moon; the density of the former being to that of the latter, as 10 to 7, and their diameters as 7930 to 2160. Ans. as 71 to 1 nearly.

21. What difference is there, in point of weight, between a block of marble containing 1 cubic foot and a half, and another of brass of the same dimensions, whose specific gravity is 8000! Ans. 496lb. 14oz.

22. What position in the line between the earth and moon, is their common centre of gravity; supposing the earth's diameter to be 7920 miles, and the moon's 2160; also the density of the former to that of the latter, as 99 to 68, or as 10 to 7 nearly, and their mean distance 30 of the earth's diameters? Ans. 633.65 miles below the surface of the earth. 23. How deep will a cube of oak sink in common water; each side of the cube being 1 foot? Ans. 11 inches. 24. How deep will a globe of oak sink in water; the diameter being 1 foot? Ans. 9.9867 inches.

25. If a cube of wood, floating in common water, have three inches of it dry above the water, and 4 inches dry when in sea-water; it is proposed to determine the magnitude of the cube, and what sort of wood it is made of?

Ans. the wood is oak, and each side 40 inches.

26. Hiero, king of Sicily, ordered his jeweller to make him a crown, containing 63 ounces of gold. The workmen thought that substituting part silver was only a proper perqui site; but Hiero, suspecting that fraud had been practised, Archimides was appointed to examine it; who on putting it into a vessel of water, found it raised the fluid 8.2245 cubic inches; and having discovered that the inch of gold more critically weighed 10.36 ounces, he found by calculation what part of the king's gold had been changed. And you are de sired to repeat the process. Ans. 28.8 ounces.

27. Supposing the cubic inch of common glass weigh 1.4921 ounces troy, the same of sea-water .59542, and of brandy .5368; then a seaman having a gallon of this liquor in a glass bottle, which weighs 3.84lb out of water, and, to conceal it from the officers of the customs, throws it overboard. It is proposed to determine, if it will sink, how much force will just buoy it up? Ans. 14.1496 ounces

28. Suppose, by measurement, it be found that a man of war, with its ordinance, rigging, and appointments, sinks so deep as to displace 50000 cubic feet of fresh water; what is the whole weight of the vessel ? Ans. 1395 tons.

29. It is required to determine what would be the height of the atmosphere, if it were every where of the same density as at the surface of the earth, when the quicksilver in the ba rometer stands at 39 inches; and also, what would be the height of a water barometer at the same time?

Ans. height of the air 291663 feet, or 5.5240 miles. height of water 35 feet.

30. If the inner axis of a hollow globe of copper, exhausted of air, be 100 feet; what thickness must it be of, that it may just float in the air? Ans. .02624 of an inch thick.

31. If a spherical baloon of copper, of of an inch thick, have its cavity of 100 feet diameter, and be filled with inflam mable air, of T of the gravity of common air, what weight will just balance it, and prevent it from rising up into the at mosphere? Ans. 21785lb.

32. Construct a quantity and show its value. (See

Book IV., Chap. II.)

ar'z
น

33. Express in a series of variables, the sym

etrical cylindrical ungulas, DBAECAB. where, AB=2ID=x, and IC=z.

Ans.

52896.89319 cubic feet, the solidity of the vacuity.

104854.11776 cubic feet, the solidity of the materials.

34. Required the superficies of a dome in the form of a right hexagonal revoloid, each side of the base being 10 feet, and height 10 feet.

35. The circumference of the base of a circular dome is 150 feet, and its height 23.873 feet; required the superficies. Ans. 3581.1 square feet.

36. If the height BC of a saloon be 3.2 feet, the BoD, of its front 4.5 feet, the distance or, of its middle part, from the arc 9 inches, and the mean circumference at m=50 feet; required the solidity of the saloon.

Ans. 138.26489 cubic feet.

C

hn D

37. What is the whole surface of a saloon round a rectangular room, the mean compass at r=67.3137 feet, the girt DrB 3.1416 feet, and the ceiling measures 16 feet long, and 12 feet broad?

Ans. 403.4727 square

feet.

38. Required the concave surface of a circular arch, raised on a rectangular base, whose sides are 27 feet 3 inches by 20 feet 4 inches.

Ans. 632.5415 square feet.