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CUBIC MEASURE

110. 1. Describe a rectangle; a square; an oblong. Draw each.

2. How many dimensions has a rectangle? a plane surface?

3. How many dimensions has a book? a block? a box? 4. Any object that has length, breadth, and thickness is called a solid. Is a book a solid? Name other solids.

5. A solid having six rectangular faces is called a rectangular solid. Is a brick a rectangular solid? Name objects which are rectangular solids.

6. A solid having six equal square surfaces is called a cube. How many edges has a cube?

7. A cube whose faces are each a foot

square is called a cubic foot.
cubic inch; a cubic yard.

Describe a

8. How many inch cubes will form a solid 12 in. long, 1 in. wide, and 1 in. thick? 12 in. long, 12 in. wide, and 1 in. thick? 12 in. long, 12 in. wide, and 2 in. thick? 12. in. long, 12 in. wide, and 3 in. thick? 12 in. long, 12 in. wide, and 6 in. thick? 12 in. long, 12 in. wide, and 12 in. thick? 9. What name may be given a solid formed by placing inch cubes 12 deep on a surface 1 ft. square? There are cu. in. in 1 cu. ft.

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RECTANGULAR
SOLIDS.

10. How many foot cubes will form a solid 3 ft. long, 3 ft. wide, and 1 ft. thick? 3 ft. long, 3 ft. wide, and 3 ft. thick?

11. How many cubic feet are there in a rectangular solid 8 ft. long, 4 ft. wide, and 4 ft. thick?

As the dimensions are all expressed in feet, the cubic contents will be found in cubic feet. In a rectangular solid having the same base, 8 ft. by 4 ft., but only 1 ft. in height, there are 1 cu. ft. × 8 × 4, or 32 cu. ft. Since the given rectangular solid is 4 ft. thick, it contains 4 times 32 cu. ft., or 128 cu. ft. 1 cu. ft. x 8 x 4 × 4 = 128 cu. ft.

When all the dimensions of a rectangular solid are expressed in like units, the contents of the solid may be found by multiplying the number of units in the length by the number of the units in the width and the product by the number of units in the thickness, and calling the result cubic units of the given dimension. Thus, the number of cubic feet in a rectangular solid 8 ft. by 4 ft. by 4 ft. is 8 x 4 x 4, or 128.

12. A pile of wood 8 ft. long, 4 ft. wide, and 4 ft. high is called a cord of wood. How many cubic feet are there in a cord of wood?

13. The number of cubic units in a solid is called its volume.

[blocks in formation]

Find the volume of rectangular solids of the following dimensions:

[blocks in formation]

7. Find the number of cubic feet of air in a room 14′

by 12' by 9'.

8. If each pupil requires 35 cu. ft. of fresh air per minute, how many cubic feet of fresh air per minute will 50 pupils require?

9. Find the number of cubic feet of air in a schoolroom 32' x 24' x 12'.

10. A watering trough is 12 ft. long, 2 ft. wide, and 2 ft. deep. How many gallons will it contain? (231 cu. in. = 1 gal.)

11. Find the number of measured bushels in a bin 6 ft. by 4 ft. by 4 ft. (2150.42 cu. in. = 1 bu.)

12. The length of a rectangular solid is 8 ft., and its width is 5 ft. If its volume is 160 cu. ft., what is its thickness?

13. How many cubic yards of dirt must be removed in excavating a cellar 36 ft. long, 24 ft. wide, and 8 ft. deep? Find the cost of making this excavation at 30 per cubic yard.

14. How many cubic yards of dirt must be removed in digging a trench for a sewer, if the trench is 3 ft. wide, 6 ft. deep, and 120 ft. long?

15. A contractor's bid for excavating a basement 60 ft. by 36 ft. and 9 ft. in depth is $216. How much is this per cubic yard?

16. How many cords of wood in a pile 24 ft. long, 4 ft. wide, and 4 ft. high?

17. How many cords of wood in a pile 8 ft. long, 2 ft. wide, and 4 ft. high?

18. At $6 per cord, find the cost of a pile of wood 20 ft. long, 4 ft. wide, and 6 ft. high.

19. State how you would find the capacity of a box car.

DIVISIBILITY OF NUMBERS

112. 1. Count by 2's to 20. Numbers that are exactly divisible by 2 are called even numbers. All numbers ending in 0, 2, 4, 6, and 8 are even numbers. Name the even numbers from 40 to 60.

2. Numbers that are not exactly divisible by 2 are called odd numbers. Name the odd numbers from 20 to 40. 3. Which of the following are even numbers: 18, 27, 31, 46, 50, 65, 123, 2456?

4. Some numbers are not exactly divisible by any whole number except themselves and 1. Such numbers are called prime numbers. Write the prime numbers below 80. Numbers that are exactly, divisible by whole numbers other than themselves and 1 are called composite numbers.

5. The factors of 15 are 3 and 5. These two numbers when multiplied together give 15. Some numbers have many factors. 2, 3, 4, 6, 8, and 12 are each a factor of 24. By what must each be multiplied to give 24?

A number which when multiplied by another number makes a given number is called a factor of the given number.

6. Name the factors of 6; of 12; of 16; of 30; of 36. Has a prime number factors?

7. The prime factors of a number are the prime numbers which when multiplied together make the number. The prime factors of 24 are 2, 2, 3, 2.

8. Name the prime factors of 18; of 20; of 36.

9. A factor is always an exact measure of a number. What numbers are exact measures of 21? of 30? of 16? of 12? of 48? Is 24 an exact measure of 48?

10. All the prime factors of a number may be found by dividing the number by one of its prime factors, and dividing each quotient in turn by one of its prime factors until the quotient is a prime number. The prime factors of 60 may be found by dividing 60 by 2, and dividing the quotient (30) by 2, and this quotient (15) by 3. The last quotient (5) is a prime number. The prime factors of 60 are the divisors, 2, 2, 3, and the last quotient, 5.

11. Find the prime factors of 48; of 72; of 80.

12. Which of the following are prime numbers: 27, 17, 39, 51, 29, 91, 53, 89, 77, 57 ?

13. Count by 5's to 35, beginning with 5. Numbers ending in 5 and in 0 are exactly divisible by 5.

14. What numbers are exactly divisible by 10? by 2? by 5?

15. Write a number the sum of whose digits is 3. Is the number exactly divisible by 3? Is 51 exactly divisible by 3?

16. Write a number the sum of whose digits is 6, 9, 12, or some other multiple of 3. Is the number exactly divisible by 3? Show by several illustrations that the following statement is correct :

A number is exactly divisible by 3 if the sum of its digits is exactly divisible by 3.

17. Which of the following are exactly divisible by 3: 54, 177, 81, 52, 819, 57, 69, 71, 213, 105, 86, 1612?

18. Write a number the sum of whose digits is divisible by 9.

A number is exactly divisible by 9 if the sum of its digits is exactly divisible by 9.

19. Which of the following are exactly divisible by 9 : 54, 504, 522, 711, 827, 218, 745, 891, 5375, 457?

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