10. Find the area of the base of a prism whose volume is 96 cu. in. and whose altitude is 15 in.

11. The radius of a cylinder is 2 in. and the volume 36 cu. in. Find the altitude.

12. A cylindrical pail is 7 in. in diameter and 6 in. deep. Does it contain a gallon?

13. The standard bushel is 18 in. in diameter and 8 in. deep. How many cubic inches does it contain?

14. The inside diameter of a water pipe is 2 in. How many gallons will flow through this pipe in one hour if the water flows at the rate of 30 ft. a minute?

15. A steel wire is of an inch in diameter. Find the weight of a mile of this wire, its specific gravity being 7.83.

16. A cylindrical cistern is 9 ft. in diameter and 10 ft. deep. How many barrels of water will it hold? (A barrel is 31 gallons. Assume 7 gal. = 1 cu. ft.)

17. A bushel of corn in the ear is approximately 21 cu. ft. How many bushels of corn in the ear can be put into a bin 24 ft. long, 7 ft. 6 in. wide, and 9 ft. deep?

18. A wagon bed is 10 ft. long, 3 ft. wide, and 26 in. deep, inside measurement. How many bushels of corn in the ear will it hold? How many bushels of wheat?

19. How many cubic yards of earth must be removed in digging a trench 4 ft. wide at the top, 2 ft. wide at the bottom, 5 ft. deep, and 10 rods long?

20. What is the weight of a column of water 3 in. in diameter and 12 ft. high?

21. In pumping water from a well there must be lifted at each stroke a column of water reaching from the level of the water in the well to the pump spout. What is the weight of the column of water to be lifted in pumping water if the diameter of the column is 3 in. and if the column is 20 ft. deep?

22. A vessel 12 in. in diameter is partly filled with water. A piece of stone of irregular shape is submersed in the water, raising the level 34 in. Find the volume of the piece of stone.

23. The end of a steel Z-bar has the dimensions and form given in Figure 50. Find the area of this end. Find the volume of such a bar 1 ft. long.

24. A steel angle has the form and dimensions given in Figure 51. Find its volume.

25. Find the volume of a piece of lead pipe 10 ft. long, 1 inch thick, and an inside diameter of 2 in. The specific gravity of the lead is 11.3. Find the weight of this pipe.

26. In making estimates, 35 cu. ft. of hard coal, or 45 cu. ft. of soft coal are called a ton. How

many tons of soft coal can be put into a bin 10 ft. square and 8 ft. deep filled to within 2 ft. of the top? How many tons of hard coal?

27. Find the volume of an iron plate of the form of Figure 52 if the plate is § in. thick.

FIG. 52

-0.5"

1.27

28. Find the volume of a steel beam 24 ft. long, a cross section of which is given in Figure 53. Find the weight of this beam if the specific gravity is 7.83.

29. Find the volume of the iron ring of Figure 54, if the outer radius is 8 in., the inner radius 4 in., and the thickness in.

30. Find the volume of this plate, FigFIG. 53 ure 55, if the diameter of the plate is 10 in., the diameter of the holes 1 in., and the thickness

in.

O

FIG. 54

FIG. 55

46. Area of the surfaces of prisms and cylinders. In many problems of painting and building it is necessary to be able to find the area of the surfaces of cylinders and prisms.

Exercise 49

1. How many faces has a rectangular solid? What is the form of these faces? What form have the faces of a cube?

2. What is the total surface of a cube the edge of which is 6 in.?

3. What is the total surface of a brick which is 2 in. by 4 in. by 8 in.?

4. Compare the surfaces and the volumes of a 4-inch cube, and a rectangular solid 2 in. by 4 in. by 8 in. What is the ratio of the volumes? Of the surfaces?

5. What is the total surface of a rectangular solid of length l, width w, and thickness t? Write a formula where S represents the surface. Solve this formula for 1, supposing S to be known.

6. Write a formula for the surface of a cube whose edge is e.

7. How many edges has a rectangular solid? What is the sum of the edges of the brick in exercise 3? Of the cube in exercise 6? Of the solid in exercise 5?

8. Cut a form like this pattern, Figure 56, from stiff paper or light cardboard. Fold along the dotted lines, paste together, and thus make a rectangular solid. What will be its dimensions?

9. In a similar way make a threeinch cube.

10. A rectangular piece of paper 12"

by 8" is rolled into

a cylinder by putting

FIG. 56

the two longer edges together. What are the circumference and the altitude of this cylinder?

11. A tin can 4 in. in diameter and 6 in. high is covered with paper. The paper is slit along the seam of the can, unrolled and spread out on a plane surface. What are the dimensions and area of the rectangle thus formed?

12. The circumference of a cylinder is c and its altitude is h. If the curved surface is rolled out into a rectangle, what are the base and altitude of the rectangle? If L is the lateral surface of the cylinder, show that L=ch. radius of the cylinder, show that L=2 rrh. mula for r; for h.

If r is the

Solve this for

13. Find the lateral surface of a cylinder 6 in. in diameter and 8 in. long.

14. How much sheet iron is required to make a piece of stove pipe 6 in. in diameter and 4 ft. long, allowing in. for the seam? 15. Cut a form like this pattern, Figure 57, from stiff paper or cardboard. Paste the edges together and thus form

a cylinder. How long is the circumference? Answer correct to the nearest in.

16. Find the total surface of the above cylinder, that is, the area of the lateral surface plus the area of the two bases.

17. The radius of a cylinder is 4 in. and its altitude is 10 in. Let L represent the lateral surface and S represent the total surface. Show that L=2XTX4X 10 sq. in., and S-2XTX 42 sq. in. +2XTX4 X 10 sq. in.

18. If r is the radius, a

the altitude, L the lateral surface, and S the total surface of

a cylinder, show that L=2 Trh Solve the last formula for h.

and that S=2 πrh+2 πr2.