172. Solids Bounded by Planes.-Planes bounding solid, as the sides of a chalk-box, are called faces and the intersections are called edges. Each corner of the solid called a vertex. Two or more are vertices. A line joining an vertex to another not in the same face, is called a diagona

EXERCISES

1. Which of the following are solids bounded by planes a brick; a house; a ball; a cracker-box; a waste-basket; & book; the teacher's desk; a piece of chalk?

2. Give illustrations of three other solids bounded by planes.

3. A chalk-box has how many faces? how many edges? how many vertices? how many diagonals? How many edges meet at each vertex?

173. Surfaces of Rectangular Solids.-A solid bounded by planes which are all rectangles, as those of the ordinary box, is called a rectangular solid. The area of the total surface of a solid is the sum of the areas of each of its faces. The area of the surface is usually spoken of as the surface. Show that the surface of a rectangular solid is

S = 2 [wl + lh + hw].

Find the surface of a chalk-bov

74. Volumes of Rectangular Solids. If the units

wn in the figure are inches, there

be as many cubic inches in the

layer as there are square inches surface in the top of the figure. e number of cubic inches in the layer times the number of layers mber of cubic inches in the volume.

will give the total

This is

(1)

(2)

which B is the area of the base. If l, w, and h are given feet, in what unit would the volume be given ?

EXERCISES

1. Measure the length of each edge of a chalk-box.

2. Find the total surface of the box.

3. Find the volume of the box.

4. The dimensions of a box are 7.45 in., 8.5 in., and 25 in. Find the total surface to two decimal places using bbreviated multiplications.

5. Find the volume of the box in Ex. 4 to two decimal laces using abbreviated multiplications.

6. A cubic foot contains about 7 gal. How many allons will a tank contain that is 18 ft. long, 1 ft. wide, nd 13 ft. deep?

7. It is estimated that a bushel of ear corn takes up 21 eu. ft. of space. How many bushels will a crib hold which s 8' by 12' by 40'?

8. A bushel of potatoes occupies about 15 cu. ft. Ho many bushels of potatoes can a commission merchant pu into a bin 6' by 10′ and 4' deep? To what depth must h pile the potatoes in this bin so as to have it contain 97 bu.

9. What is the value of the potatoes in a bin 8' by 12 and 6' 8" deep at $1.45 per bushel?

10. The altitude of a rectangular solid is 12.35 dm. and its volume is 45.75 dm3. Find the area of its base.

Solve equation (2), page 159 for h; for B.

12. The base of a rectangular box is 45 cm2. Find it height if the volume is 134 cm3.; 563.72 cm3.; 463.48 cc.

=

13. If V decimal places.

3456 cu. in. and h

=

215 in., find B to two

14. A cord of wood is 4' by 4' by 8'. How many cords are there in a pile of wood 16' long and 6′ high in which the sticks are 4' long?

15. A schoolroom is supposed to have 240 cu. ft. of air for each occupant. Does a schoolroom 20′ by 24′ and 12′ high seating 30 pupils fulfil this requirement?

16.

Measure the length, width, and height of your schoolroom and compute its volume. Does it fulfil the requirement for Ex. 15?

17. Construct an open rectangular box with a base 13 in. by 2 in. and 3 in. high.

For construction work, use a heavy paper which will fold over easily without breaking. Make very carefully all figures asked for and save them for future use.

18. Construct an open rectangular box which will have

75. Diagonals of Rectangular Solids. The figure shows -t the diagonal AD is the hypotenuse

the right triangle ACD. Hence,

oking at the top base, it is seen that

C is the hypotenuse of the right tri- D

y substituting this value of AC2 in (1) we get

B

A.

(2)

1. Find the length of the diagonal of the rectangular olid, the edges of which are 3 cm., 4 cm., and 12 cm.

2. Make the measurements and compute the length of a diagonal in your schoolroom.

3. Compute the length of the diagonals of the box constructed for Ex. 17, page 160, by using the lengths of its edges. To verify the computations use a thin stick or knitting-needle to measure the length of the diagonal.

4. In the same way find the diagonals for the box constructed for Ex. 18 on page 160.

5. Find the diagonal of a cube whose edges are 1 in.; 6 cm.; 4.5 in.

6. Note that the diagonal of a cube is 3 times the edge of the cube. Express this as an equation.

7. Express by an equation the total surface, S, of a cube

176. Lumber Measurements.-Lumber is measured by "board feet," usually called feet. A board foot has a surface of one square foot and a thickness of one inch or less. A board 10' long, 12" wide, and 1" or less thick contains 10 board feet; a board 10' long, 6′′ wide, and 1′′ or less thick contains 5 board feet. A piece of lumber 12′ long, 12′′ wide, and 2′′ thick contains 24 board feet. These would be written: 1" x 12" x 10'; 1" x 6" x 10'; 2" x 12" x 12'. Which dimension comes first? last? The number of board feet in a piece of lumber equals its length in feet times its width in feet times its thickness in inches. If l, w, and t are expressed in feet and fractions of a foot, then

Show that the equation and the statement above correspond.

Flooring is usually 3", 4", or 6′′ wide. The groove uses up" so the width that really makes the floor is only 21", 31", and 51′′ wide. Carpenters usually add of the floor space for waste.

All lumber comes in even foot lengths, as 8', 10', up to 22'. The longer lengths and the thicker lumber costs the more per board foot. These must be sawed from larger logs of which there are a less number. Clear fine-grained lumber is also more expensive than is coarse lumber containing knots.

177. Lumber Quotations.-The price of lumber is quoted on 1000 board feet. Thus, $ 30 per M or $45 per M. What is the meaning of M? Sometimes lumber is also quoted on 100 board feet, as $3 or $4.50 per 100 feet.

1//

Laths are 1" x 11" x 4′ and are sold by the bundle of 50 laths in each bundle.

The price of shingles is quoted per M or 4 bunches. See

170