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Denominate Numbers - Lumber.

344.— A foot of lumber is a board 1 foot square and 1 inch thick or its equivalent.

NOTE 1.—An exception to the foregoing is made in the measurement of boards less than 1 inch in thickness. A square foot of such boards is regarded as a foot of lumber whatever the thickness.

Observe that a piece of board 1 inch wide, 1 inch thick, and 12 feet long, is 1 foot of lumber.

EXERCISE. Tell the number of feet of lumber in each of the following boards, the thickness in each case being 1 inch (or less):

1 in. wide and 12 ft. long. 2 in. wide and 12 ft. long. 3 in. wide and 12 ft. long. 4 in. wide and 12 ft. long. 7 in. wide and 12 ft. long. 13 in. wide and 12 ft. long. 9 in. wide and 12 ft. long. 12 in. wide and 12 ft. long. (a) How many feet (of lumber) in the eight boards?

PROBLEMS.

1. How much lumber in 6, 12-ft. 1-in. boards whose widths are 11 in., 13 in., 9 in., 10 in., 12 in., and 14 in. ?

2. How much lumber in 5, 12-ft. {-in. boards whose widths are 10 in., 12 in., 12 in., 11 in., and 14 in. ?

3. How much lumber in 7, 12-ft. -in. boards whose widths are 9 in., 8 in., 5 in., 7 in., 8 in., 6 in., and 9 in, ?

4. How much lumber in 8, 12-ft. 1-in. boards each of which is 12 inches wide ?

5. How much lumber in 54, 12-ft. 1-in. boards each of which is 6 inches wide ?

(b) Find the sum of the five results.

Denominate Numbers-Lumber.

PROBLEMS.

NOTE 2.— A 14-foot board contains į more lumber than a 12-ft. board of the same width and thickness. Hence to find the number of feet of lumber in 14-foot rds, find the number of feet in as many 12-foot boards * and add to the result å of itself.

1. How much lumber in 6, 14-ft. 1-in. boards whose widths are 11 in., 12 in., 12 in., 15 in., 10 in., and 13 in. ?

2. How much lumber in a pile of 14-ft. boards whose united width is 8 feet 7 inches ?

3. How much lumber in 56, 14-ft. boards each of which is 6 inches wide ? +

4. How much lumber in 24, 14-ft. boards each of which is 12 inches wide ?

(a) Find the sum of the four results.

PROBLEMS.

NOTE 3.— A 16-foot board contains } more lumber than a 12-foot board of the same width and thickness. Make a rule for finding the number of feet of lumber in 16-foot boards.

1. How much lumber in 6, 16-ft. 1-in. boards whose widths are 12 in., 10 in., 14 in., 13 in., 12 in., and 10 in. ?

2. How much lumber in a pile of 16-foot boards whose united width is 9 feet 8 inches ?

3. How much lumber in 48, 16-ft. boards each of which is 6 inches wide ?

4. How much lumber in 34, 16-ft. boards each of which is 12 inches wide ?

(b) Find the sum of the four results.

* Take the nearest integral number of feet.
| How much lumber in 1 14-foot board 6 inches wide ?
| How much lumber in 1 14-foot board 12 inches wide ?

Denominate Numbers — Lumber.

PROBLEMS.

NOTE 4. — A 14-inch board contains more lumber than a 1-inch board of the same width and length. A 11-inch board contains } more lumber than a 1-inch board of the same width and length.

1. How much lumber in 4, 12-foot 14-in. boards whose widths are 12 in., 13 in., 14 in., and 13 in. ?

2. How much lumber in 4, 16-foot 11-in. boards whose widths are 13 in., 16 in., 12 in., and 13 in. ?

3. How much lumber in 4, 18-foot 1-in. boards, each of which is 12 inches wide ?

4. How much lumber in 4, 16-ft. 11-in. boards, each of which is 6 inches wide ?

(a) Find the sum of the four results.

PROBLEMS. NOTE 5. A “2 by 4, 12” is a piece of lumber 2 in. thick, 4 in. wide, and 12 feet long.

Find the number of feet of lumber in each of the following items:

1. 16 pieces 2 x 4, 12.
2. 18 pieces 4 x 4, 12.
3. 25 pieces 2 x 8, 12.
4. 30 pieces 2 x6, 12.
5. 20 pieces 4x 6, 12.

6. 32 pieces 6 6, 12.
(b) Find the sum of the six results.

Observe that in a 12-foot piece of lumber there are as many feet as there are square inches in the cross-section. A piece of lumber 1 in. by 1 in. and 12 feet long is 1 foot of lumber; a piece 2 in. by 2 in. is 4 feet of lumber; a piece 2 in. by 3 in. is 6 feet of lumber, etc.

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NOTE 6. - In the measurement of timbers of all sizes it is customary to consider each piece as containing the integral number of feet nearest to the actual content. Thus, a piece of 2 x 4, 14, actually contains 9f feet, but in all lumber yards it is counted as 9 feet. A piece of 2 x 4, 16, actually contains 10% feet, but it is counted as 11 feet.

Find the number of feet of lumber in each of the following items :

1. 16 pieces 2 x 4, 14.*
2. 24 pieces 4 x 4, 14.
3. 32 pieces 2 x 8, 14.
4. 17 pieces 4 * 6, 14.
5. 15 pieces 8 x 8, 16.
6. 12 pieces 4 x 10, 16.
7. 14 pieces 8 x 12, 16.

8. 6 pieces 12 x 12, 24.
(a) Find the sum of the eight results.

PROBLEMS.

NOTE 7.—“ Lumber at $ 15 per M,” means that the lumber is sold
at the rate of $ 15 per 1000 feet.
Find the cost :

1. 26, 16-foot, 6-in. fence boards @ $15 per M.
2. 34, 14-foot 12-in. stock boards “ $18 per M.
3. 20 pieces 2 x 4, 16,

“ $16 per M.
4. 14 pieces 4x6, 18,

“ $16 per M.
5. 25 pieces 4 x 6, 16,

$15
per

M.
6. 18 pieces 4 x 4, 14,

per

M. (b) Find the sum of the six results.

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* Find the number of feet in 1 piece of 2 X 4, 14 (nearest whole number of feet) and multiply by 16.

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Geometry.

345. To find the solid content of a cylinder or of a right prism.*

Observe that in any cylinder or right prism the number of cubic units in one layer 1 unit high (as indicated in the diagrams) is equal to the number of square units in the area of the base. Thus, if there are 41 square units in the area of the base there are 41 cubic units in one layer. The content of the entire solid is as many times the cubic units in one layer, as the solid is linear units in height. Hence the rule as usually given: Multiply the area of the base by the altitude.

TO THE TEACHER. — This rule must be carefully interpreted by the pupil. He must not be allowed the misconception that area multiplied by any number can give solid content, except through such interpretation as is suggested in the above observation.

PROBLEMS.

1. Find the solid content of a square right prism whose base is 6 in. by 6 in., and whose altitude is 8 inches.

2. Find the approximate solid content of a cylinder 6 inches in diameter and 10 inches long.

* A right prism is a solid whose bases, or ends, are similar, equal, and parallel plane polygons, and whose lateral faces are perpendicular to its bases.

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