2. Required the solidity of a cone whose altitude is 27 feet, and the diameter of the base 10 feet.

Ans. 706,86 cubic feet.

3. Required the solidity of a cone whose altitude is 10 feet, and the circumference of its base 9 feet?

Ans. 22,5609 cubic feet.

4. What is the solidity of a cone, the diameter of whose base is 18 inches, and altitude 15 feet?

Ans. 8,83575 cubic feet.

5. The circumference of the base of a cone is 40 feet, and the altitude 50 feet: what is the solidity?

Ans. 2122,1333 solid feet.

PROBLEM V.

5. To find the surface of the frustum of a cone.

QUEST.-5. How do you find the convex surface of the frustum of a cone? How do you find the entire surface ?

Mensuration of the Round Bodies.

RULE.

Add together the circumferences of the two bases, and multiply the sum by half the slant height of the frustum; the product will be the convex surface, to which add the areas of the bases, when the entire surface is required.

EXAMPLES. 3

1. What is the convex surface of

the frustum of a cone, of which the

slant height is 12

feet, and the cir

cumferences of the base 8,4 and 6

feet.

We merely take the sum of the circumferences of the bases, and multiply by half the slant height, or side.

Operation.

8.4

6

14,4

half side 6,25

area 90 sq. ft.

2. What is the entire surface of the frustum of a cone, the side being 16 feet, and the radii of the bases 2 and

3 feet?

Ans. 292,1688 sq. ft.

Mensuration of the Round Bodies.

3. What is the convex surface of the frustum of a cone, the circumference of the greater base being 30 feet, and of the less 10 feet; the slant height being 20 feet?

Ans. 400 sq. ft.

4. Required the entire surface of the frustum of a cone whose slant height is 20 feet, and the diameters of the bases 8 and 4 feet.

Ans. 439,824 sq. ft.

PROBLEM VI.

6. To find the solidity of the frustum of a cone.

RULE.

1st. Add together the areas of the two ends and a geometrical mean between them.

2d. Multiply this sum by one-third of the altitude and the product will be the solidity.

EXAMPLES.

1. How many cubic feet in the frustum of a cone whose altitude is 26 feet, and the diameters of the bases 22 and 18 feet?

First, 22x,7854-380,134-area of lower base:

QUEST.-6. How do you find the solidity of the frustum of a cone?

Mensuration of the Round Bodies.

and 18,7854-254,47-area of upper base. Then, √380,134 × 254,47=311,018=mean.

26

Then, (380,134+254,47+311,018) × -8195,39

3

which is the solidity.

2. How many cubic feet in a piece of round timber the diameter of the greater end being 18 inches, and that of the less 9 inches, and the length 14,25 feet? Ans. 14,68943.

3. What is the solidity of the frustum of a cone, the altitude being 18, the diameter of the lower base 8 and that of the upper base 4?

Ans. 527,7888.

4. What is the solidity of the frustum of a cone, the altitude being 25, the circumference of the lower base 20, and that of the upper base 10?

Ans. 464,216.

5. If a cask, which is composed of two equal conic frustums joined together at their larger bases, have its bung diameter 28 inches, the head diameter 20 inches, and the length 40 inches, how many gallons of wine will it contain, there being 231 cubic inches in a gallon? Ans. 79,0613.

Mensuration of the Round Bodies.

PROBLEM VII.

7. To find the surface of a sphere.

RULE.

Multiply the circumference of a great circle by the di ameter, and the product will be the surface.

2. What is the surface of a sphere whose diameter

is 24 ?

Ans. 1809,5616.

3. Required the surface of a sphere whose diameter is 7921 miles.

Ans. 197111024 sq. miles.

4. What is the surface of a sphere the circumference of whose great circle is 78,54?

Ans. 1963,5.

QUEST.-7. How do you find the surface of a sphere?