2. The slant height of the frustum of a square pyramid is 25 feet, the side of one base is 12 feet, and of the other 4 feet; what is the whole surface?

3. The slant height of the surface of the frustum of a cone is 20 feet, the circumferences of the bases being 30 feet and 15 feet respectively; what is the whole surface?

To find the volume of the frustum of a pyramid

or cone.

RULE.

To four times the area of a middle section add the areas of the ends, and multiply by one-sixth the altitude.

Note. By this rule the contents of nearly all regular solids and their frustums can be correctly ascertained. Let pupils practice on the preceding examples in solids, and test its accuracy.

1. The length of a mast is 60 feet, the diameter in the middle is 1 foot, and the diameters of the ends are 6 and 18 inches respectively; what is the number of cubic feet in the mast?

ANALYSIS.-The area of a middle section is 12 x .7854= .7854 ft.; the area of the two ends=(1)2+(1)2× .7854=1.9635 ft.; .7854 x 43.1416 ft. four times the area of the middle section; 3.1416+1.9635=5.1051; 5.1051 × 60=51.051 cu. ft.

=

2. The frustum of a square pyramid of marble is 9 feet in height, the side of the lower base is 12 feet, the side of the upper base is 2 feet, the middle section measures 7 feet on each side; what is the weight of the frustum?

3. A lump of gold in the form of the frustum of a cone measures 8 inches in diameter at one end, and 10 inches at the other, its length being 1 foot; what is the value of the lump, if an ounce is worth $20.69?

THE SPHERE.

A Sphere is a solid bounded by a curved surface, every part of which is equally distant from a point within it, called the centre.

D

B

The Diameter of a sphere is a line which passes through the centre and is terminated by the surface. The diameter is frequently called the Axis.

The Radius of a sphere is a line drawn from the centre to any part of the surface.

To find the surface of a sphere.

RULE.

Multiply the diameter by the circumference.

1. The diameter of a globe is 10 inches; what is the convex surface?

10 x 3.1416 × 10 = 314.16 sq. in., convex surface.

2. The circumference of a sphere is 1000 feet; what is the convex surface?

3. The diameter of the earth is about 7960 miles; what is its surface?

4. If the radius of a globe is 25 feet, what is the surface?

To find the volume of a sphere.

RULE.

Multiply the cube of the diameter by .5236.

1. What is the solidity of a globe whose diameter is 100 feet?

1003 × .5236 = 1000000 × .5236 = 523600 cubic feet, volume.

2. What is the solidity of the earth, if its diameter is 7960 miles?

3. The radius of a ball is 14 inches; what will it cost to gild it, at the rate of 20 cents a foot, and what will it weigh if it is made of walnut?

4. What is the weight of a cannon-ball of cast iron, 12 inches in diameter?

REVIEW PROBLEMS.

1. What is the height of a room whose shape is that of a cube, the solid contents being 3375 cubic feet?

ANALYSIS. As the solidity is the product of three equal factors, the cube root of 3375 ft., or 15 ft., is the height required. 2. A cubical bin holds 500 bushels; what is the length of one of its inside edges?

3. A mill hopper in the form of the frustum of a square pyramid measures 3 in. on each side at the bottom, and 4 feet at the top, its height being 18 inches; how many bushels of wheat will it hold?

4. If a cubic foot of air at the level of the sea weighs .0768 lb., what is the weight of air which covers one mile square of the ocean, the thickness being reckoned as 1 foot?

5. What is the difference in the weight of two vessels in the form of cylinders, one filled with milk and the other with water, each measuring 1 ft. in diameter and 10 in. in height, the empty vessels weighing 1 pound each?

6. How much more will an iron cannon-ball 10 inches in diameter weigh, than one 5 inches in diameter?

7. How many spheres of gold 1 inch in diameter are equal in weight to a sphere of the same metal, 6 inches in diameter?

8. The area of the base of a cylinder that will contain 1000 gallons, is 10 sq. ft.; what is the altitude?

9. The altitude of a pyramid is 9 ft. and the volume. is 192 cu. ft.; what is the side of the square base?

10. The volume of a cone is 36 cu. in.; the area of the base is 6 sq. in.; what is the altitude?

11. The surface of a square prism, excluding the bases, is 144 sq. ft.; the solidity of the prism is 216 cu. ft.; what is the length of one of its sides?

12. The dimensions of a coal-bin are 16 ft. by 8 ft. at the base, and 28 ft. by 10 ft. at the top; the altitude is 6 ft.; how many tons of 2240 lb. each, will the bin hold?

13. What will be the cost of digging a cellar 30 ft. long, 12 ft. wide, and 6 ft. deep, a cubic yard being called a load, and the cost of each load being 50 cents?

GAUGING.

Gauging is the process of finding the capacity of a cask, barrel, or other vessel.

The usual method of obtaining the capacity of a barrel is to reduce its dimensions to those of an equivalent cylinder by finding its mean diameter, and apply the rule for the cylinder.

The best method of obtaining the contents of any cask or barrel is by weight; the weight of the empty barrel being marked upon it, and the weight of a cubic foot of the liquid being known.

To find the mean diameter of a cask.

RULE.

Add to the head diameter .5, .55, .6, .65, or .7 of the difference between the bung and head diameters, according to the curvature of the cask. Thus, if the difference of the diameters is but 1 or 2 inches, add .5 of the difference; if the difference is 3 or 4 inches, add .55, and so on.

To find the contents of a cask in gallons.

RULE.

Multiply the square of the mean diameter in inches by the length in inches, and this product by .0034.

Note. The multiplier .0034 is the quotient obtained by dividing the factor .7854 by 231, the number of cubic inches in a gallon.

1. How many gallons in a cask whose head diameter is 24 inches, bung diameter 26 inches, and length 40 inches?

ANALYSIS.-26-24-2. 24+ (2 × .5) = 25 inches, the mean diameter of a cylinder equal in area to the cask. 252 x 40 x .003485 gal.; or, 252 × .7854 × 40 = 19635, the area of the cylinder, in inches, and 19635÷231, the number of inches in a gallon, gives 85 gallons.

2. What is the number of gallons in a barrel of vinegar, the weight of the empty barrel being 55 pounds, and the entire weight 269 pounds, a gallon of vinegar weighing 83 pounds?

ANALYSIS.-269 lb.-55 lb.=214 lb., the weight of the vinegar; 214÷83=2416, the number of gallons required.

3. Find the contents of a cask of common alcohol by both the preceding methods, the head diameter being 28 inches, the bung diameter 34 inches, and the length 40 inches; the weight of the empty cask being 68 lb., and the entire weight 1006.25 lb.

4. A barrel of pure alcohol weighs 258.43 lb.; the empty barrel weighs 50 lb.; what is the number of gallons in the barrel?

5. A barrel of linseed oil weighs 290.62 lb.; the empty barrel weighs 55 lb.; what is the number of gallons in the barrel?