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Observe: 1. That the dimensions of the pyramid and of the prism are the same, and that those of the cone and of the cylinder are the same. 2. That the volume of the pyramid is that of the prism, and the volume of the cone is that of the cylinder.

By geometry, it is shown that the volume of a pyramid is of that of a prism having an equal base and an equal altitude. Hence,

The volume of a pyramid equals one third of the volume of a prism of like dimensions.

The volume of a cone equals one third of the volume of a cylinder of like dimensions.

But we have already learned that the volume of a prism or of a cylinder is found by multiplying its unit of measure by the product of the area of its base by its altitude. Hence,

The volume of a pyramid or of a cone is found by multiplying its unit of measure by one third the product of the altitude and the area of the base.

1. Find the volume of a cone whose altitude is 12 in. and the diameter of the base 8 in.

z. How often can a conical cup 8 in. high and 6 in. in diameter be filled from a cylindrical vessel 2 ft. high and 6 in. in diameter?

3. Find the volume of a pyramid whose base is 12 in. square and whose altitude is 30 in.

4. A square pyramid whose side is 18 in. is 32 in. high. Find its volume.

5. Find the volume of a pyramid whose altitude is 12 ft. and whose base is a square 8 ft. on a side.

6. Find the contents of a rectangular pyramid 15 ft. high, the sides of whose base are 10 ft. and 12 ft. respectively.

7. A pile of grain in the form of a cone is 15 ft. in diameter and 6 ft. high. How many bushels of grain does it contain?

8. A concrete mixer, 6 ft. from base to apex, being conical in forın, and measuring 3 feet across the base, is filled six times an hour. How many cubic feet of concrete material may be manufactured with it in a week of six working days of 8 hours each?

9. A wooden hopper supplying coal to a furnace is in the form of an inverted pyramid. If it is 8 ft. deep and 6 ft. square at the top, how many tons of hard coal will it contain? 10. A square pyramid, the perimeter of whose base measures 64 inches, contains 2048 cubic inches. Find its altitude. 11. The contents of a cone are 471.24 cu. ft.; the altitude is 18 ft. Find the diameter.

Examine the figure:

SPHERES

Observe: 1. That the solids formed by the dissected part of the sphere are pyramids.

2. That the radius of the sphere is

the altitude of the pyramids.

3. That the combined bases of the pyramids form the convex surface of the sphere.

The volume of a sphere is found by multiplying its unit of measure by one third the product of the radius and its convex surface.

It is also shown by geometry that

The volume of a sphere equals four thirds of the cube of the radius multiplied by 3.1416, or (representing the radius by r and 3.1416 by π) 1⁄2 πÃ3.

Find the volume of :

1. A globe 12 inches in diameter.

2. A bowling ball with a radius of 4 inches.

3. A cannon ball with a diameter of 8.2 inches.

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Comparative Volumes of Cone, Sphere, and Cylinder

Доп

[graphic]

Compare the diameters of the bases and the altitudes of the cone and the cylinder with each other, and with the diameter of the sphere. Observe that the dimensions are all equal.

By geometry it is shown that the volumes of these three solids are in the ratio of 1, 2, and 3. The volume of the cone is, and of the sphere of that of the cylinder.

SIMILAR SURFACES

Similar figures are plane surfaces that have exactly the same shape, but differ in size. Point out similar figures:

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D

A

2"

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Find the areas of the similar surfaces on p. 300. Square the corresponding lines of the similar figures and express their ratio. Compare the ratio of the areas of the similar figures with the ratio of the squares of their corresponding lines. In the similar figures observe:

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the squares of their corresponding lines; that is, 6 sq. in.: 24 sq. in. as 22:42, or as 32: 62.

Corresponding lines of similar plane surfaces are proportional. The areas of similar plane figures are proportional to the squares of their corresponding lines.

Written Work

1. If a rectangle is 20 ft. by 50 ft., what will be the length of a similar rectangle 30 ft. in width?

2. The side of a square field is 40 rods. Find the side of a similar field that contains four times as many acres.

3. A lady buys two rugs, one 6 ft. by 9 ft. and a similar rug 18 ft. in length. Find its width.

4. In East Park a circular fountain is 40 ft. across and in West Park a circular fountain is 26 ft. across. The area of the first fountain is how many times the area of the second fountain?

2ft.

4ft.

5. Find the length of the side marked x in the larger of these similar triangles.

4ft.

X ft.

6. It costs $19.50 to gild a sphere 18 inches in diame

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Similar solids are solids that have the same shape but differ in contents or volume; thus,

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Observe: 1. That the length of the first solid is to the length of the second as 1:2, and that the heights and widths of the solids are in the same ratio.

2. That the ratio of their contents or volume equals the ratio of the cubes of their corresponding lines; that is, 2 cu. in.: 16 cu. in. as 13:28; or as 23:48.

The corresponding lines of similar solids are proportional.

The contents or volumes of similar solids are proportional to the cubes of their corresponding lines.

Written Work

1. Compare in volume a 5-in. cube with a 10-in. cube.

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