EXAMPLES FOR PRACTICE.

1. The radius of a circle containing 28.2744 sq. ft., is 6 ft.; what is the radius of a circle containing 175.7150 sq. ft.?

28.2744 175.7150 62: () quired. Hence, 225 15, Ans.

=

225, square of radius re

2. If it cost $75 to inclose a circular pond containing a certain area, how much will it cost at the same rate to inclose another, containing 5 times the area of the first? Ans. $167.70.

3. If a cistern 6 feet in diameter hold 80 barrels of water, what must be the diameter of a cistern of the same depth to hold 1200 barrels ? 23,23+

4. If a pipe 1.5 in. in diameter will fill a cistern in 5 h., what must be the diameter of a pipe that will fill the same cistern in 55 min. 6 sec.? Ans. 3.5 in.

PROBLEM V.

693. To find the side of a cube, the solid contents being given.

NOTE. This case, arithmetically considered, requires us to separate a number into three equal factors.

The solid contents of a cube are found by cubing the length of one side; hence,

RULE. Extract the cube root of the given contents.

EXAMPLES FOR PRACTICE.

1. What must be the length of the side of a cubical bin that shall contain the same quantity as one that is 24 ft. long, 18 ft. wide, and 4 ft. deep? Ans. 12 ft.

2. What must be the length of the side of a cubical bin that will contain 150 bushels?

3. What must be the depth of a cubical cistern that will hold 200 bbl. of water? 1:3

4. How many sq. ft. in the surface of a cube whose solidity is 79507 cu. ft.?

3

Ans. 11094.

PROBLEM VI.

694. To find the three dimensions of a parallelopiped, the solid contents and the ratio of the dimensions being given.

NOTE 1. This case, arithmetically considered, requires us to separate a number into three factors, proportional to three given numbers.

The three dimensions will be like multiples of the proportional terms, (691); the product of the three dimensions, or the solid contents, will therefore contain the product of the three proportional terms, and the cube of the common ratio which the proportional terms respectively bear to the corresponding dimensions, and no other factor. Hence the

RULE. I. Divide the given contents by the product of the terms proportional to the three dimensions, and extract the cube root of the quotient.

II. Multiply the root thus obtained by each proportional term; the products will be the corresponding sides.

NOTE 2. -The dimensions are supposed to be taken in a direction perpendicular to the faces of a solid, and to each other.

EXAMPLES FOR PRACTICE.

1. A pile of bricks in the form of a parallelopiped contains 3000 cu. ft, and the length, breadth, and thickness, are to each other as 4, 3, and 2, respectively; what are the dimensions of the pile? Ans. 10, 15, and 20 ft.

2. Three numbers are to each other as 2, 5, and 7, and their continued product is 4480; required the numbers.

Ans. 8, 20, and 28. 3. Separate 100 into three factors which shall be to each other as 2, 2, and 3. Ans. 3.76414+; 4.70518 +; 5.64622 -.

4. A person wishes to construct a bin that shall be of equal width and depth, and the length three times the width, and that shall contain 450 bushels of grain? what must be its dimensions?

PROMISCUOUS EXAMPLES.

1. There is a park containing an area of 10 A. 2 R. 20 P., and the breadth is equal to of the length. If two men start from one corner and travel at the rate of 3 miles per hour, one going by the walk around the park, and the other taking the diagonal path through the park, how much sooner will the latter reach the opposite corner than the former? Ans. 1 min. 5 sec. 2 S.3 2. What is the length of one side of a square piece of land containing 40 acres? Ans. 69.28+ rd.

3. The ground situated between two parallel streets is laid out into equal rectangular lots whose front measure is 44 per cent. greater than the depth. Now, if the streets were 20 feet further apart, the ground could be laid out into square lots of the same area as the rectangular. What is the distance between the streets? Ans. 100 feet.

4. How much less will it cost to fence 40 acres of land in the form of a square, than in the form of a rectangle of which the breadth is the length, the price per rod being $1.40?

Ans. $112.

5. If a cistern 6 feet in diameter holds 80 barrels of water, how much water will be contained in a cistern of the same depth and 18 feet in diameter?

6. What is the length of the side of a square which contains the same area as a role 7 feet in diameter? Ans. 6.204+ feet. 7. What is the length of the side of a square which can just be inclosed within a circle 42 inches in diameter?

Ans. 29.7-in.

8. If it costs $75 to inclose a circular fish pond containing 3 A. 86 P., how much will it cost to inclose another containing 17 A. 110 P.? Ans. $167.70.

NOTE 1.-It is proved in Geometry that all similar solids are to each other as the cubes of their like dimensions. Hence, any dimension may be found by proportion, when its ratio to the corresponding dimension of a known similar solid is given.

9. What is the length of the side of a cubical vessel that contains as much as one whose side is 6 ft.?

Ans. 3 ft.

24

10. How many globes 4 in. in diameter are equal in volume to one 12 in. in diameter?

9.42+

11. If an ox that weighs 900 lb. girt 6.5, what is the weight of an ox that girts 8 ft.? Ans. 1677 lb. 14+ oz. 12. If a cable 3 in. in circumference supports a weight of 1500 lb., what must be the circumference of a cable that will support 4960 lb.? 92

13. If a stack of hay 4 feet high contain 4 tons, how high must a similar stack be to contain 20 tons ?

SERIES.

695. A Series is a succession of numbers so related to each other, that each number in the succession may be formed in the same manner, from one or more preceding numbers. Thus, any number in the succession, 2, 5, 8, 11, 14, is formed by adding 3 to the preceding number. Hence, 2, 5, 8, 11, 14, is a series.

696. The Law of a Series is the constant relation existing between two or more terms of the series. Thus, in the series, 3, 7, 11, 15, we observe that each term after the first is greater than the preceding term by 4; this constant relation between the terms is the law of this series.

The law of a series, and the number of terms on which it depends being given, any number of terms of the series can be formed. Thus, let 64 be a term of a series whose law is, that each term is four times the preceding term. The term following 64 is 64 × 4, the next term 64 × 42, etc.; the term preceding 64 is 64 ÷ 4. Hence the series, as far as formed, is 16, 64, 256, 1024.

697. A series is either Ascending, or Descending, according as each term is greater or less than the preceding term. Thus, 2, 6, 10, 14, is an ascending series; 32, 16, 8, 4, is a descending series.

698. An Extreme is either the first or last term of a series. Thus, in the series, 4, 7, 10, 13, the first extreme is 4, the last, 13.

699. A Mean is any term between the two extremes. Thus, in the series, 5, 10, 20, 40, 80, the means are 10, 20, and 40.

700. An Arithmetical or Equidifferent Progression is a series whose law of formation is a common difference. Thus, in the arithmetical progression, 3, 7, 11, 15, 19, each term is formed from the preceding by adding the common difference, 4.

701. An arithmetical progression is an ascending or descending series, according as each term is formed from the preceding term by adding or subtracting the common difference. Thus, the ascending series, 7, 10, 13, 16, etc., is an arithmetical progression in which the common difference, 3, is constantly added to form each succeeding term; and the descending series, 20, 17, 14, 11, 8, 5, 2, is an arithmetical progression in which the common difference is constantly subtracted, to form each succeeding term.

702. A Geometrical Progression is a series whose law of formation is a common multiplier. Thus, in the geometrical progression, 3, 6, 12, 24, 48, each term is formed by multiplying the preceding term by the common multiplier, 2.

703. A geometrical progression is an ascending or descending series, according as the common multiplier is a whole number or a fraction. Thus, the ascending series, 1, 2, 4, 8, 16, etc., is a geometrical progression in which the common multiplier is 2; and the descending series, 32, 16, 8, 4, 2, 1, 1, §, etc., is a geometrical progression in which the common multiplier is 2.

704. The Ratio in a geometrical progression is the common multiplier.

705. In the solution of problems in Arithmetical or Geometrical progression, five parts or elements are concerned, viz:

The conditions of a problem in progression may be such as to require any one of the five parts from any three of the four remaining parts; hence, in either Arithmetical or Geometrical Progression, there are 5 × 4 20 cases, or classes of problems, and no more, requiring each a different solution.