2. What is the content of the frustum of a parabolic spindle, whose length is 18, greatest diameter 18, and least 10 ? Ans. 3404 23776.

NOTE. The solidities of the hyperboloid and hyperbolic spindle are to be found by Rule to Prob. XXIV. And those of their frustums by Prob. XXV; where some examples of them are given.

MISCELLANEOUS QUESTIONS

IN MENSURATION OF SUPERFICIES AND SOLIDS.

1. WHAT difference is there between a floor 28 feer

long by 20 broad, and two others, each of half the dimen sions; and what do the three floors come to at 45s. per 100 square feet?

Ans. Diff. 280 square feet. Amount 18 guineas,

2. An elm plank is 14 feet 3 inches long, and it is desired, that just a square yard may be slit off from it; at what distance from the edge must the line be struck?

Ans. 7 inches.

3. A ceiling contains 114 yards 6 feet of plastering, and the room is 28 feet broad; what is the length of it?

Ans. 369 feet.

4. A common joist is 7 inches deep and 24 thick; but a scantling just as big again, that shall be three inches thick, is wanted; what will the other dimension be?

Ans. 11 inches.

5. A wooden trough cost 3s. 6d. for painting within at 6d. per yard; the length of it was 102 inches, and the depth 21 inches; what was the width?

Ans. 334 inches.

6. If a court yard be 47 feet 9 inches square, and a foot path of 4 feet wide be laid with purbeck stone along one side of it; what will the paving of the rest with flints come to, at 6d. per square yard?

Ans. 51. 16s. Old.

7. There are two columns in the ruins of Persepolis left standing upright; one is 64 feet above the plane, and the other 50; in a straight line between these stands an ancient small statue, the head of which is 97 feet from the summit of the higher, and 86 feet from the top of the lower column, the base of which measures just 76 feet to the centre of the figure's base; required the distance between the tops of the two columns.

Ans. 152 feet nearly.

8. The perambulator, or surveying wheel, is so contrived, as to turn just twice in the length of a pole, or 16 feet; required the diameter.

Ans. 2'626 feet.

9. In turning a one horse chaise within a ring of a certain diameter, it was observed, that the outer wheel made two revolutions while the inner made but one; the wheels were both 4 feet high; and, supposing them fixed at the statutable distance of 5 feet asunder on the axletree, what was the circumference of the track described by the outer wheel? Ans. 63 feet nearly,

10. What is the side of that equilateral triangle, whose area cost as much for paving at 8d. sterling a foot, as the pallisading of the three sides did at a guinea a yard ? Ans. 72'746 feet.

11. A roof, which is 24 feet 8 inches by 14 feet 6 inches, is to be covered with lead, at 8lb, to the square foot; what will it come to, at 18s. per cwt. ?

Ans. 221. 19s. 101d.

12. Having a rectangular marble slab, 58 inches by 27, I would have a square foot cut off parallel to the shorter edge; I would then have the like quantity divided from the remainder parallel to the longer side; and this alternately repeated, till there shall not be the quantity of a foot left; what will be the dimensions of the remaining piece?

Ans. 20'7 inches by 6'086.

13. If from a right-angled triangle, whose base is 12 and perpendicular 16 feet, a line be drawn parallel to the perpendicular cutting off a triangle, whose area is 24 square feet; required the sides of this triangle,

Ans. 6, 8, and 10,

14. If a round pillar, 7 inches over, have 4 feet of stone in it; of what diameter is the column of equal length, that contains 10 times as much?

Ans. 22'136 inches.

15. The area of an equilateral triangle, whose base falls on the diameter, and its vertex in the middle of the arc of a semicircle, is equal to 100; what is the diameter of the semicircle?

Ans. 26'32148.

16. It is required to find the thickness of the lead in a pipe of an inch and a quarter bore, which weighs 14lb. per yard in length; the cubic foot of lead weighing 11325 ounAns. '20737 inches.

ces.

17. What is the length of a chord, which cuts off of the area of a circle, whose diameter is 289 ?

Ans. 278 6716.

18. What will the diameter of a globe be, when the solidity and superficial content are expressed by the same number? Ans. 6.

19. A sack, that will hold 3 bushels of corn, is 224 inches broad, when empty; what will that sack contain, which, being of the same length, has twice its breadth or circumference ? : Ans. 12 bushels.

20. A carpenter is to put an oaken curb to a round well, at 3d. per foot square; the breadth of the curb is to be 74 inches, and the diameter within 13 feet; what will be the expense? Ans. 5s. 24d.

21. A gentleman has a garden 100 feet long and 80 feet broad, and a gravel walk is to be made of an equal width half round it; what must the breadth of the walk be to take up just half the ground? Ans. 25'968 feet.

22. Seven men bought a grinding stone of 60 inches diameter, each paying part of the expense; what part of the diameter must each grind down for his share?

Ans. The 1st, 4'4508,

2d, 4'8400,

3d, 5'3535,

4th, 6'0765,

5th, 7*2079,

6th, 9'3935,

7th, 22 6778.

23. A maltster has a kiln, that is 16 feet 6 inchés square; but he wants to pull it down and build a new

one, that will dry 3 times as much at once as the old one; what must be the length of its sides?

Ans. 28 feet 7 inches.

24. How many 3 inch cubes may be cut out of a 12 inch cube? Ans. 64.

25. What will the painting of a conical spire come to, at ed. per yard; supposing the height to be 118 feet, and the circumference of the base 64 feet? Ans. 141. 83d.

26. The diameter of a standard corn bushel is 18 inches, and its depth 8 inches; what must the diameter of that bushel be, whose depth is 7 inches?

Ans. 19'1067.

27. To divide a cone into three equal parts by sections parallel to the base, and to find the altitudes of the three parts; the height of the whole cone being 20 inches.

Ans. The upper part 13'867.

The middle part 3'604.

The lower part 2 ̊528.

28. A gentleman has a bowling green 300 feet long and 200 feet broad, which he would raise one foot higher by means of the earth to be dug out of a ditch, that goes round it; to what depth must the ditch be dug, supposing its breadth to be every where 8 feet?

Ans. 7.

29. How high above the earth must a person be raised, that he may see of its surface?

Ans. To the height of the earth's diameter.

30. A cubic foot of brass is to be drawn into a wire of of an inch in diameter; what will the length of the wire be, allowing no loss in the metal?

Ans. 97784 797 yards, or 55 miles 984 797 yards.