3. What is the value in currency of 7,984 ounces of silver bullion at $19.8 per pound?

4. What must be paid for a $200 four-and-a-half per cent bond, in order to make it an 8 per cent investment?

5. G and H engaged in business as partners. G invested $10,000 and H $5,000, G sharing and H of the gains and losses. Their assets at the close of the year consisted of cash, $2,100; bills receivable, $4,400; merchandise, $13,000; and personal accounts, $8,000, 10 per cent of which are considered worthless. Their liabilities are bills payable, $1,625; personal accounts, $5,625. G drew out during the year $850 and H $1,075.

If H should retire from the firm, how much would he be entitled to receive?

CUBE ROOT.

1155. To cube a number is to employ it three times as a factor.

The cube of 4, written 43, is 4 × 4 × 4, or 64.

Find the cube of 1, 9, 6, 3, 5, 8, 2, 7.

To find the cube root of a number is to find one of the three equal factors of the number.

The cube root of 343, written 343, is 7.

The cube of 25, 20+ 5, is equal to the following:

We have seen (Art. 1031) that

(20 + 5)2 = 202 + 2 × 20 × 5 + 52

(20+ 5)3 = 203 + 3 × 202 × 5 + 3 × 20 × 52 + 53

which may be written in this way,

203 + [(3 × 202) + (3 × 20 × 5) + 52] × 5.

1,200 as a trial divisor, the second number is seen to be 6 or less.

Taking 5 as the second number, we add to the 1,200 three times the product of the first and second (300), and the square of the second (25), making a total of 1,525. Multiplying this sum by the second number, we get 7,625, which is equal to the difference between 15,625 and 8,000. The second number is, therefore, 5, and the cube root of 15,625 is 25.

In the last example we point off three places, beginning at the right, and find the greatest cube in the first period, placing its cube root as the first figure of the answer.

VOLUME OF SPHERE.

1158. Cut up a sphere (a round potato, for instance) into a number of small pieces, passing the knife in each case through the center of the sphere.

Each piece is a solid, having for its base a portion of the surface of the sphere, and for its altitude the radius of the sphere.

When the pieces become very numerous, the base of each may be considered a plane, and the solid a pyramid. The volume of each pyramid is

equal to the base altitude; and the total volume of all, which is the volume of the sphere, is equal to the total surface of all the bases, which is the surface of the sphere, multiplied by altitude, that is, radius.

therefore,

Surface of sphere = 4πR2,

volume of sphere = 4π R2 × R= }πR3.

1159. Slate Exercises.

1. Find the volume of a sphere whose radius is 3 inches. 2. If the diameter of a sphere is 3 inches, what is its volume? 3. What is the ratio between the volumes of two spheres whose diameters are 1 foot and 2 feet, respectively?

4. Find the ratio between the volume of a sphere 1 foot in diameter, and that of a cube whose side is 1 foot.

5. The radius of a sphere is 18 inches. What is the circumference of a great circle? The surface? The volume?

6. What is the weight of an iron cannon-ball 12 inches in diameter, considering the weight of a cubic foot of water as 1,000 ounces, and considering iron 7.5 times as heavy as water?

7. Find the ratio between the volume of a sphere 4 inches in diameter, and that of a cylinder 4 inches in altitude, radius of base 4 inches.

NOTE. - Indicate the volume of each, and cancel.

8. A man has a cubical block of hard wood, its side measuring one foot, which he wishes made into a sphere one foot in diameter. What decimal part of the block is cut away?

The volume of the sphere is about what fraction of the volume of the cube?

9. How much would be cut away in making a cylinder one foot in diameter and one foot high, from the above cubical block? About what fraction of the volume of the cube is the volume of the cylinder?

1160. Chicago Public Schools.

1. A general wished to remove 80,000 pounds of provisions from a fortress in 9 days. It was found that in 6 days 18 men had carried away but 18 tons. How many men would be required to carry away the remainder in 3 days?

2. A, B, and C enter into partnership. A puts in $500 for 4 months, B $400 for 6 months, and C $800 for 3 months; they gain $340. Find each man's share of the gain.

3. A school-room is 40 feet long, 30 feet wide, and 14 feet high. Find the difference between the length of a diagonal drawn on the floor and one drawn from the floor to the ceiling.

4. A merchant bought $6,500 worth of goods for cash, and sold them on 4 months' credit at 15% advance. He got the note discounted at 6% to pay the bill. How much did he make?

5. A merchant in Galveston paid $4,265 for a draft on St. Louis at 30 days' sight, exchange being 31% premium, interest 8%. Required the face of the draft.

6. Find the solid contents and the surface of a sphere 12 inches in diameter.

7. A man traveled from New York, lon. 74° 3', to San Francisco, lon. 122° 27', without changing his watch. On arriving at San Francisco was his watch too fast or too slow, and how much?

8. Find the entire surface of a cone whose altitude is 10 feet, and diameter of base 8 feet.

9. A, having a quantity of canal stock, sold 25% of it to B, who sold 33% of his purchase to C; who sold 37% of his purchase to D, who received 6 shares. How many shares had A at first?

10. A and B engaged in business for one year with the same capital. A increased his capital by of it, and B lost of his. The difference between their capitals then was $1,040. Find the capital of each at the beginning.

11. The number of copies in the first edition of the "Lady of the Lake" was 2,050, and was to the number in the second edition as 41 to 69. Find the number in the second edition.

12. At the end of 12 months A, B, and C, having a joint capital of $6,000, find that they have lost $625. A's capital of $2,500 has been in the business for 12 months, B's of $1,500 for 8 months, and C's of $2,000 for 4 months. Divide the loss among them.

13. Find the proceeds of the following note:

$1,050,000.

CHICAGO, Feb. 13, 1885.

Six months after date I promise to pay to the order of Geo. Hall, One Thousand Fifty Dollars, with interest at 6 per cent.

Discounted at 8 per cent May 13.

HENRY SHAW,