6. Two observers at A and B in the same vertical plane with an airplane at C observe it at angles of 50° and 60°, respectively. The observers are 440 yards apart. Make a drawing to scale and find the height of the airplane. 7. A hostile ship

at S is observed from two forts A and B 2 miles apart. The ZA is 75°, and ZB is 60°. The guns of each fort have a range of 3 miles. Is the ship within range of either fort?

B

FIG. 41

8. A ladder 20 ft. long is set against a building. When the ladder is safe, the angle between the ladder and the building is not less than 15°. What is the highest point that can safely be reached by the top of the ladder?

CHAPTER V

VOLUMES AND SURFACES OF SOLIDS

42. Comparison of rectangular solids. This block of wood with square corners is 6 inches long, 2 inches wide, and 3 inches high.

A solid of this shape is called a rectangular solid.

FIG. 42

Into how many blocks each 2"x2"x3" can this block be cut?

A box is 6"x4"X4" (inside measurement). How many blocks each 3"x2"x2" can be laid along one edge of the bottom if laid with the long edge of the block along the long edge of the box? How many such rows will it take to cover the bottom? How many such layers will it take to fill the box? How many blocks are required to fill the box?

Another box is 6"x4"x4". How many blocks each 2"x2"X1" can be laid along one edge of the bottom of this box? How many blocks are required for one layer on the bottom? How many layers to fill the box? How many blocks to fill the box?

A rectangular solid whose length, breadth, and thickness are all equal is called a cube.

A cube each of whose dimensions is one inch is called an inch cube, or a cubic inch. A cube each of whose dimensions is 3 inches is called a three-inch cube, and so on.

Draw a picture of a nine-inch cube, using an eighth of an inch for an inch in measuring the nearest edge.

How many three-inch cubes can be laid along one edge of the bottom of a hollow nine-inch cube? How many threeinch cubes does it take to cover the bottom? How many to fill the nine-inch cube? In the same way find how many four-inch cubes in an eight-inch cube.

FIG. 43

-3".

A rectangular solid is usually measured by com

puting the number of cubes of a certain size it contains. If the dimensions of the solid are measured in inches, the cubic inch is used as the unit of measure.

By the volume of a solid is meant the number of cubes it contains whose edge is the unit of length with which the dimensions of the solid are measured. Thus, the volume of a solid 2'X3'x5' means the number of cubic feet it contains.

43. Board measure. In measuring lumber it is not convenient to use a cube as the unit of measure. Instead, a solid 1 foot long, 1 foot wide

and 1 inch thick is used. This unit is called a board foot.

How many board feet in a board 1 inch thick, 1 foot wide, and 2 feet long? In

FIG. 44

a board 1 foot wide, 1 inch thick, and 5 feet long? In a board 2 feet wide, 1 inch thick, and 3 feet long?

How many board feet in a board 1 inch thick, of a foot wide, and 2 feet long? One inch thick, 6 inches wide, and 6 feet long?

It is usual to speak of a number of feet of lumber when a number of board feet is meant.

How many feet of lumber in a piece 2 inches thick, 1 foot wide, and 10 feet long?

Make a rule for computing the number of feet in a piece of lumber. This rule should mention the units in which the length, width, and thickness are measured.

Make a formula from this rule, calling the length 1, the width w, and the thickness t, and assuming that l and w are numbers of feet while t is a number of inches.

Rule. To find the number of board feet in a piece of lumber multiply together the number of feet in the length, the number of feet in the width, and the number of inches in the thickness.

The length of a board is usually given in feet, while both the width and thickness are given in inches. To reduce the width to feet it is only necessary to divide by 12, hence the following

Rule. To find the number of board feet in a board multiply together its length in feet, its width in inches, and its thickness in in hes and divide by 12.

These processes should first be indicated, then cancellation used.

EXAMPLE. How many board feet in a board 15′8′′×2′′?

In measuring lumber in board feet boards less than one inch thick are counted one inch thick.

Thus, a board 4" wide, 12' long, and " thick is counted as 4X12X1 board feet, or 4 board feet.

12

Exercise 45

1. A brick is 8"x4" x2". How many bricks will it take to fill a box 2′×1′×1'?

2. How many bricks can be loaded in a wagon bed 8'x2' 8" x 10"?

3. Making no allowance for mortar, how many bricks are required for a foundation 22′ ×3′×16′′?

4. How many board feet in a board 3′ long, 1' wide, and 2" thick?

5. How many feet in a piece of lumber 3'x6"×2"?

6. How many feet in a two-by-four 8' long? (A two-byfour is 2" thick and 4" wide.)

7. How many feet in a four-by-four 16' long?

8. How many feet of one-half-inch flooring are required for a floor 18'×24′, making no allowance for waste?

9. The lumber bill for a certain house calls for 78 pieces 2"x4" X9′ and 24 pieces 2" ×8′′ ×12′. How much does this lumber cost at 44 cents a foot?

10. The backstop for a baseball diamond is 10' high and 16' long. It is built of six-inch boards, 1" thick and 16' long. Three posts are used, each 4′′ ×4′′ ×16'. How many feet of lumber are required?

11. How many feet in a wagon load of lumber 12' long 3' 4" wide, and 2' 6" high if the boards are a half-inch thick?

12. How many feet of lumber are required to make a covered box whose outside dimensions are 16"x26"×45", using 1-inch lumber and allowing of a foot waste in squaring? How much will this lumber cost at $65 a thousand?

13. Lumbermen use the formula, V

(B+b)l

=

2

for finding

the number of cubic feet in a log, where B and b are the areas of the ends of the log in square feet, and I is its length in feet. Find the number of cubic feet in a log whose ends are 16 in. and 18 in. in diameter and whose length is 12 ft.