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Exercises. Miscellaneous Exercises

1. If a cube and a sphere have equal volumes, what is the ratio of the radius of the sphere to the edge of the cube?

2. Given that the diagonal of a cube is 4 √3 in., find the radius of the sphere whose area is equal to that of the cube.

3. The radius of the base of a right circular cylinder is r and the altitude of the cylinder is h. Find the radius and the volume of a sphere whose area is equivalent to the lateral surface of the cylinder.

4. If the area of a zone of one base is n times the area of the circle which forms this base, the altitude of the zone is equal to the diameter of the sphere multiplied by (n−1)/n. Discuss the special case in which n

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1.

5. Find to the nearest 0.1 in. how far from the center of a sphere of radius 8 in. a plane should be passed so as to cut from the sphere a circle 154 sq. in. in area.

6. Find the ratio of the volume of a sphere to the volume of the inscribed cube.

7. Consider Ex. 6 for the circumscribed cube.

8. Find the ratio of the volume of the cube inscribed in a sphere to that of the cube circumscribed about the sphere. 9. Find the difference between the volumes of two cubes, one inscribed in a sphere of radius 1 in. and the other circumscribed about it.

10. Find the difference between the volume of a frustum of a pyramid and the volume of a prism each 20 ft. high, given that the bases of the frustum are squares 20 ft. and 12 ft. respectively on a side, and the base of the prism is the section of the frustum parallel to the bases and midway between them.

Omit Ex. 10 if § 125 was not taken.

11. In certain parts of the United States, stacks of hay are shaped roughly like a barn, as shown in this figure. The farmers use this rule for finding the approximate number of tons: Take the "overthrow" (the length ABCDE); subtract the width; divide by 2, and call the result the height. Multiply this height by the product of the length and width, and call the result the cubic contents. Divide this by 412 to find the number of tons of wild hay; by 450 for timothy; and by 512 for alfalfa or clover. Consider the accuracy of the rule for finding the volume in the case here shown.

13'

B

E

18'

A

12. Using the rule and the dimensions of the figure in Ex. 11, find the approximate number of tons in a stack of wild hay of timothy; of clover.

13. The square of the diagonal of a cube is how many times the square of an edge?

14. Find the ratio of the sum of the squares of all the edges of a cube to the sum of the squares of all the diagonals of the faces; of all the diagonals of the cube itself.

15. Find the length of the perpendicular from a vertex of a regular octahedron of edge e to the plane determined by the four adjacent vertices.

16. The common perpendicular between two opposite edges of a regular tetrahedron is equal to half the diagonal of the square constructed on an edge.

17. The sum of the squares of the edges of any tetrahedron is four times the sum of the squares of all the lines joining the midpoints of opposite edges.

18. Find the volume of the regular tetrahedron of which the sum of the areas of the faces is 4f.

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19. The six planes that pass through the six edges of a tetrahedron and bisect the respective opposite edges meet in a point.

20. A cubic foot of copper is drawn into a wire 2000 ft. long. Find the diameter of the wire.

21. Find the volume of a pyramid with equal lateral edges e and a base which is an equilateral triangle of side s. 22. Consider Ex. 21 for the case in which the base is a regular octagon of side s.

23. The base of a regular pyramid of volume V and height h is a square. Find the length of a lateral edge.

24. In a cube a plane passes through the midpoints of three edges that meet in the same vertex. Given that the volume of the cube is e3, find the volume of the tetrahedron thus cut off.

25. An iron casting is in the form of a right circular cone upon the base of which is a hemisphere of the same radius outside the cone. If the casting, which is 7 in. in diameter and 9 in. long, is placed in a cylindric can 8 in. in diameter and 10 in. high, filled with water, how much water remains in the can?

26. Water is flowing into a tank through a pipe 2.1 in. in diameter at the rate of 3 ft. (linear) per second. Allowing 231 cu. in. to a gallon, how much water will flow into the tank in 1 hr.?

27. If the centers of two intersecting spheres are 5 in. apart, and the radii of the spheres are 3 in. and 4 in. respectively, what is the area of the circle formed by their intersection?

28. A sphere 1 ft. in diameter is cut from a cube of lead 1 ft. on an edge. If the pieces cut off are melted and cast into another sphere, what is the diameter of this sphere?

SUPPLEMENT

I. INCOMMENSURABLE CASE OF $108

231. Supplementary Work. In the study of geometry there are many topics that might be taken in addition to those found in any textbook. The theorems already given are those which are looked upon as fundamental, and are usually required as preliminary to more advanced work. These propositions and a reasonable number of originals selected from the exercises will be all that most classes have time to consider. It occasionally happens, however, that a class is able to do more than this, and then more exercises may be selected from the large number supplied, and a few additional topics may be studied. For this latter purpose the supplementary work is added, but its study should not be undertaken at the expense of the fundamental propositions and exercises. The work on practical mensuration (§§ 259-266), however, may be taken with advantage in place of some of the less important propositions, and all students are advised to read the brief sketch of the history of geometry (§§ 268–274) for general information.

232. Incommensurable Magnitudes. Two magnitudes of the same kind which cannot both be expressed in integers in terms of a common unit are called incommensurabl magnitudes.

The diagonal and the side of a square are, as the student already knows, incommensurable lines. We also have incommensurable numbers such as 2 and √3, for there is no number which is contained in both of these numbers without a remainder.

233. Volume of a Rectangular Parallelepiped. Referring to the case discussed in $ 108, let us suppose that the edges of the rectangular parallelepiped are all incommensurable.

Let U, in the figure below, be the unit of volume and let u be the unit of length. Then if u is applied to AB as many times as possible, there is a remainder r, less than u. Similarly, there is a remainder r on the width BE, and a remainder r, on the height AD.

Now if we let the unit U decrease indefinitely, r1, r2, and r, also de- h crease indefinitely; that is, as U→0,

D

then u→0, r1→0, r2→0, and A

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G

F

X

-W

U

Since r1, r2, and r, all approach zero as a limit, we see that BZ→l, BW→w, and BX→h, as U is taken continually smaller and smaller.

Let P be the volume of the rectangular parallelepiped with the dimensions 1, w, h, and let P' be the volume of the one with the dimensions BZ, BW, BX.

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By extending the principles of limits (§ 28), we may assume, as seems evidently to be the case, that

BZ BW BX →lwh.

But

P'=BZ BW · BX.

.. P=lwh.

§ 108

§ 28, 2

No proof of this case is satisfactory for a textbook of this type. If rigorous, the proof is too difficult for an elementary class; if simple, it lacks scientific accuracy. The fact that the elementary proofs often given are open to serious scientific criticism has led most careful writers to outline merely the general nature of the proof as has been done above. Teachers are advised to require only that the above discussion be read understandingly.

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