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Denominate Numbers.

336. FARM PROBLEMS.

Find how many acres in

1. A piece of land 1 rod by 160 rods.
2. A piece of land 7 rods by 160 rods.
3. A piece of land 13 rods by 160 rods.
4. A piece of land 22 feet * by 160 rods.
5. A piece of land 87 yards * by 160 rods.

(a) Find the sum of the five results.
6. A piece of land 8 rods by 80 rods.
7. A piece of land 17 rods by 80 rods.
8. A piece of land 377 rods by 80 rods.
9. A piece of land 618 feet * by 80 rods.
10. A piece of land 550 yards * by 80 rods.

(b) Find the sum of the five results.

11. A piece of land 12 rods by 40 rods. 12. A piece of land 27 rods by 40 rods. 13. A piece of land 46 rods by 20 rods. 14. A piece of land 36 rods by 20 rods. 15. A piece of land 264 feet * by 20 rods.

(c) Find the sum of the five results. 16. A piece of land 1 rod by 1 mile. 17. A piece of land 11 rods by 1 mile. 18. A piece of land 66 feet * by 1 mile. 19. A piece of land 99 yards * by 1 mile. 20. A piece of land 198 feet * by 1 of a mile.

(d) Find the sum of the five results. 21. A piece of land of a mile long and as wide as the schoolroom.

* Change to rods.

Denominate Numbers.

337. FARM PROBLEMS.

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1. A piece of land 1 foot wide and 43560 feet long is how many acres ?

2. Change 43560 feet to miles.

3. A piece of land 1 foot wide must be how many miles in length to contain 1 acre? 4. Some country roads are 66 feet wide.

How many acres in 81 miles of such road?

5. How many acres in 1 mile of road that is 4 rods wide ?

6. A farmer walking behind a plow that makes a furrow 1 foot wide will travel how far in plowing 1 acre?

7. A farmer walking behind a plow that makes a furrow 16 inches wide will travel how far in plowing 1 acre?

8. If a mowing machine cuts a swath that averages 4 feet in width how far does it move in cutting 1 acre?

9. If potatoes are planted in rows that are 3 feet apart (a) how many miles of row to each acre? (b) How many rods of row to each acre ? (c) If 4 rods of row on the average yield 1 bushel, what is the yield per acre?

10. Strawberry plants are set in rows that are 2 feet apart. (a) How many miles of row to the acre ? (b) How many rods of row to the acre? (c) How many feet of row to the acre ?

11. If corn is planted in rows 33 feet apart and if the “hills are 34 feet apart in the row, how many hills to each

acre ? *

* Think of each

hill” as occupying a piece of land 34 ft. by 34 ft.

Geometry.

338. TO FIND THE AREA OF A CIRCLE.

1. Cut one half of a circular piece of paper as indicated in the diagram.

Observe that if the circle is cut into a very large number of parts and opened as shown in the figure, the circumference of the circle becomes, practically, a straight line.

Note.-Imagine the circle cut into an infinite number of parts and thus opened and the circumference to be a straight line.

Observe that a circle may be regarded as made up of an infinite number of triangles whose united bases equal the circumference and whose altitude equals the radius. Hence to find the area of a circle we have the following:

RULE I. Multiply the circumference by 1 of the diameter.

2. It has already been stated that if the diameter of a circle is 1, its circumference is 3.141592. * Hence the area of a circle whose diameter is 1 is (3.141592 x ) .785398.

3. A circle whose diameter is 2, is 4 times as large as a circle whose diameter is 1 ; a circle whose diameter is 3, 9 times as large, etc.

Hence to find the area of a circle we have also the following:

RULE II. Multiply the square of the diameter by .785398. 4. The approximate area may be found by taking : (or .78) of the square of the diameter. See Werner Arithmetic, Book II., page 256.

* See page 229, note.

339. MISCELLANEOUS PROBLEMS. *

1. Find the approximate area of a circle whose diameter is 20 feet.

2. What is the area of a circle whose diameter is 1 foot ? 1 yard? 1 rod ? 1 mile?

3. What is the area of a circle whose diameter is 2 feet? 2 yards ? 2 rods ? 2 miles ?

4. A horse is so fastened with a rope halter that he can feed over a circle 40 feet in diameter. Does he feed over more or less than 5

square

rods ? 5. Find the approximate length (in rods) of the side of a square containing 2 acres.

6. Find the approximate diameter (in rods) of a circle whose circumference is one mile.

7. Find the approximate area of the circle described in problem 6.

8. Find the approximate circumference of a circle whose diameter is 30 rods.

9. The expression "a bicycle geared to 68” means that the machine is so geared that it will move forward at each revolution of the pedal shaft as far as a 68-inch wheel would move forward at one revolution. How far does a bicycle geared to 68” move forward at each revolution of the pedal shaft? A bicycle "geared to 70"?

10. What is the approximate circumference of the largest circle that can be drawn on the floor of a room 40 ft. by 40 ft. if at its nearest points the circumference is 2 feet from the edge of the floor?

*Require the pupil to make an estimate of the answer to each problem before attempting to solve it with the aid of a pencil.

DENOMINATE NUMBERS.

VOLUME MEASURE.

340. The standard unit of volume measure is a cubic yard which is the equivalent of a 1-yard cube. This unit, like the cubic foot and the cubic inch, is derived from the corresponding unit of linear measure.

CUBIC MEASURE.

1728 cubic inches (cu. in.)
27 cubic feet

1 cubic foot (cu. ft.).
- 1 cubic yard (cu.yd.).

EXERCISE. 1. Show by a drawing that there are 27 cu. ft. in a 1-yard cube.

2. How many cubic inches in 1 half of a cubic foot ?
3. How many cubic inches in a 1-foot cube?
4. How many cubic feet in 1 third of a cubic yard ?
5. How many cubic feet in a t-yard cube?

6. Estimate in cubic feet the amount of air in the school room.

7. Estimate in cubic yards the amount of air in the school room.

8. Estimate in cubic inches the capacity of your dinner box.

9. Estimate in cubic feet the capacity of some wagon box. 10. Estimate in cubic inches the volume of the school globe. *

* A globe is a little more than of the smallest cube from which it could have been made. See Werner Arithmetic, Book II., p. 266.

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