EXAMPLES FOR PRACTICE.

339. Solve the following by compound proportion:

1. If 12 men dig a trench 40 rods long in 24 days of 10 hours each, how many rods can 16 men dig in 18 days of 9 hours each?

Ans. 36 rods. 2. If a piece of iron 7 ft. long, 4 in. wide, and 6 in. thick weighs 600 lb., how much will a piece of iron weigh that is 16 ft. long, 8 in. wide and 4 in. thick? Ans. 1,828 lb.

3. If 24 men can build a wall 72 rods long, 6 feet wide, and 5 feet high in 60 days of 10 hours each, how many days will it take 32 men to build a wall 96 rods long, 4 feet wide and 8 feet high, working 8 hours a day? Ans. 80 days.

4. The horsepower of an engine varies as the mean effective pressure, as the piston speed and as the square of the diameter of the cylinder. If an engine having a cylinder 14 inches in diameter develops 112 horsepower when the mean effective pressure is 48 pounds per square inch and the piston speed is 500 feet per minute, what horsepower will another engine develop if the cylinder is 16 inches in diameter, piston speed is 600 feet per minute, and mean effective pressure is 56 pounds per square inch? Ans. 204.8 horsepower. Referring to the example in Art. 335, what will be the volume of a cylinder 20 inches in diameter and 24 inches long?

5.

Ans. 7,539.84 cubic inches. 6. Knowing that the product of 3×5×7 × 9 is 945, what is the product of 6 X 15 × 14 × 367 Ans. 45,360.

7. The speed in miles per hour of a locomotive is directly proportional to the diameter of its driving wheels and the number of revolutions they make in one minute. A locomotive with driving wheels 66 inches in diameter runs 29.45 miles in an hour when the wheels make 150 revolutions per minute; how many miles will be run in one hour by a locomotive having wheels 72 inches in diameter running 220 revolutions per minute? Ans. 47.12 miles per hour.

8. The capacity of a cylindrical boiler is proportional to its length and the square of its diameter. A boiler 12 feet long and 48 inches in diameter will hold 1,128 gallons; what is the capacity of a boiler 16 feet long and 42 inches in diameter ? Ans. 1,151.5 gallons.

9. The power that may be transmitted by a belt is proportional to its width and the diameter and number of revolutions made by the pulley on which it runs. If a belt 12 inches wide will transmit 10 horsepower when running over a pulley 20 inches in diameter that makes 125 revolutions per minute, how many horsepower may be transmitted by a belt 8 inches wide when running over a pulley 30 inches in diameter that makes 200 revolutions per minute?

Ans. 16 horsepower.

10. The load that a beam supported at the two ends will carry is directly proportional to its width and the square of its depth, and inversely proportional to its length. If an oak beam 8 inches wide, 12 inches deep, and 15 feet long will safely carry a load of 13,824 pounds, what is the safe load that a beam 10 inches wide, 16 inches deep, and 20 feet long will support? Ans. 23,040 pounds.

MENSURATION AND USE OF LETTERS

IN FORMULAS.

FORMULAS.

340. The term formula, as used in mathematics and in technical books, may be defined as a rule in which symbols are used instead of words; in fact, a formula may be regarded as a shorthand method of expressing a rule. Any formula can be expressed in words, and when so expressed it becomes a rule.

Formulas are much more convenient than rules; they show at a glance all the operations that are to be performed; they do not require to be read three or four times, as is the case with most rules, to enable one to understand their meaning; they take up much less space, both in the printed book and in one's note-book, than rules; in short, whenever a rule can be expressed as a formula, the formula is to be preferred.

As the term "quantity" is a very convenient one to use, we will define it. In mathematics, the word quantity is applied to anything that it is desired to subject to the ordinary operations of addition, subtraction, multiplication, etc., when we do not wish to be more specific and state exactly what the thing is. Thus, we can say "two or more numbers," or "two or more quantities;" the word quantity is more general in its meaning than the word number.

341. The signs used in formulas are the ordinary signs indicative of operations, and the signs of aggregation. All these signs are explained in arithmetic, but some of them will here be explained in order to refresh the student's memory.

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