PROBLEM XXVII.

To find the solidity of the frustum of a paraboloid.

RULE

Multiply the sum of the squares of the diameters of the two ends by the height, and the product by 3927, for

the content.

EXAMPLES.

1. Required the content of the paraboloidal frustum ABCD, the diameter AB being 20, the diameter DC 40, and the height EF 22.

2. What is the content of the frustum of a paraboloid, the greatest diameter being 30, the least 24, and the altitude 9 ?

Ans. 5216-6266.

PROBLEM

PROBLEM XXVIII.

To find the solidity of a parabolic spindle.

RULE..

Multiply the square of the middle or greatest diameter by the length, and the product by 41888, for the content.

EXAMPLES.

1. Required the content of the parabolic spindle ACBD, whose length AB is 40, and the greatest diameter CD 16.

2. What is the solidity of a parabolic spindle, whose length is 18, and its middle diameter 6 feet?

Ans. 271'4336.

PROBLEM

VOL. II.

S

PROBLEM XXIX.

To find the solidity of the middle frustum of a parabolic spindle.

RULE.

Add together 8 times the square of the greatest diameter, 3 times the square of the least diameter, and 4 times the product of the two diameters; multiply the sum by the length, and the product by '05236, for the solidity.

EXAMPLES.

1. Required the content of the frustum of a parabolic spindle EGHF, the length AB being 20, the greatest diameter CD 16, and the least diameter EF 12.

0

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 feet long by 20 broad, and two others, each of half the dimensions; 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. 36 feet.

4. A common joist is 7 inches deep and 2 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. 274 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 fints come to, at 6d. per square yard?

Ans. 51. 16s. ožd

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