14. Use the formula given in the previous problem and find the number of cubic feet in each of the following logs: 15. The Doyle Rule for finding the number of board feet in a log is the one most widely used in the United States. If N is the number of board feet in a log, d the diameter of the small end in inches, and 7 the length in feet, then by the Doyle Rule, N= 7. Find by this rule the number =(d–4)%. of board feet in a log if the diameter of the small end is 18 in. and the length 16 ft. 16. Use the Doyle Rule and find the board feet in logs with the following dimensions: How many layers? How many cubic inches in the solid? From the answers to these questions we see that the volume of the solid is 3×4×5 cu. in. =60 cu. in. Answer similar questions concerning a solid 6′′ long, 5′′ wide, and 4" high; also about a solid 6×41′′×6′′; also about a solid 8′′×51′′×91′′. Exercise 46 1. The volume of the solid 3"x4"x5" may be found by taking the product 5×4×3 cu. in. Point out a part of the solid in Figure 45 whose volume is 3 cu. in. Point out a part whose volume is 4×3 cu. in. 2. The volume of this solid may also be found by taking the product 3X4X5 cu. in. Point out a part of the solid whose volume is 5 cu. in. Point out a part whose volume is 4×5 cu. in. Find the volumes of rectangular solids with the following dimensions: 9. 4.3", 9.6", 11.1", correct to .01 cu. in. 10. 25.1', 14.3', 75.8', correct to .01 cu. ft. 11. 8.3', 5.25', 7.4', correct to .01 cu. ft. 12. 22′, 22', 4' 8", correct to .1 cu. yd. 13. A lot 100 ft. by 60 ft. is to be filled in with earth to a depth of 1 ft. How many loads will be required if a load is 1 cu. yd.? 14. A cellar is 20 ft. long, 18 ft. wide, and 8 ft. deep. How many cubic yards of earth must be removed? What is the cost of excavating at 50¢ a cubic yard? 15. A flat car is 40 ft. 4 in. long, 8 ft. 6 in. wide, and 3 ft. 6 in. deep. How many cubic yards of gravel will it hold if it is level full? 16. The inside dimensions of a concrete watering trough How many gallons will it hold? are 8′ 6′′ by 2' 6" by 1' 6". Allow 7 gallons to i cu. ft. 17. The outside dimensions of the trough of the preceding problem are 9'x3'x2'. How many cubic feet of concrete are needed to make it? 18. An ice house is 30 ft. long, 18 ft. wide, and 15 ft. high. How many cubic feet of ice will it hold, allowing 1 ft. on the sides and ends and 2 ft. above and 2 ft. below for sawdust? 19. A grain car is 30 ft. long and 8 ft. 6 in. wide. It is filled to a depth of 5 ft. with wheat. How many bushels of wheat does it then contain? Find the answer both by using 2150.4 cu. in. for one bushel, and by using cu. ft. for one bushel. What is the difference between the two answers? 20. This car is filled through a chute the end of which is 8 in. square. The wheat runs through the chute at the rate of 10 ft. a second. How long will be required to fill the car to the depth of 5 ft.? 21. There are 30 pupils in a room 32 ft. long, 24 ft. wide, and 14 ft. high. How many cubic feet of air in the room for each pupil? 22. Fresh air is sent into this room through a ventilator shaft that is 1'x1' 6". The air moves through the ventilator shaft at the rate of 10 ft. a second. How many cubic feet of air come into the room in one second? How many cubic feet of air come into the room for each pupil per minute? Ice is frozen over 23. The surface of a pond is 2 acres. it to a depth of 8 in. A cubic foot of water weighs 62 lb., and the specific gravity of the ice is .92. How many tons of ice on the pond? Answer to the nearest ton. 24. The average annual rainfall in Central Illinois is 38 in. How many gallons is that per square foot? 45. Volumes of prisms and cylinders. The surfaces that bound a solid are called its faces. Prism A prism is a solid two of whose faces are equal and parallel polygons and whose other faces are parallelograms. The equal and parallel faces are called the bases of the prism. The other faces of the prism are called the lateral surfaces. Point out prisms in the school room. Figure 47 is an example of a cylinder. The bases are equal The altitude of a prism or and parallel circles. a cylinder is the perpendicular distance between the bases. Point out cylinders in the school room. Cylinder FIG. 47 If the lateral surface of a prism or a cylinder is perpendicular to the bases, the prism or cylinder is called a right prism or a right cylinder. Exercise 47 1. If the base of this prism, Figure 48, contains 12 sq. in., what is the volume of a portion of it that is 1 in. high? What is the volume of the prism if it is 10 in. high? How can you find the volume of a right prism if you know the area of the base and the altitude? FIG. 48 2. The radius of the base of this cylinder is 4 in. What is the area of the base? What is the volume of a portion of this cylinder that is 1 in. high? What is the volume of the cylinder if the altitude is 6 in.? What is the formula for finding the area of a circle when the radius is given? Make a rule for finding the volume of a cylinder when the radius and altitude are given. 3. Point out examples of prisms in the schoolroom and tell how many faces each has. 4. Point out solids that are not prisms or cylinders, and state why they are not. 5. Do you see any solids that are bounded partly by plane surfaces and partly by curved surfaces? FIG. 49 Rule for the volume of a prism. The volume of a prism equals the product of the base by the altitude. Formula. V=bh. Rule for the volume of a cylinder. The volume of a cylinder equals π times the square of the radius times the altitude. Formula. V =πr2h. Exercise 48 Find the volumes of cylinders with the following radii and altitudes: 1. Radius 3", altitude 10", correct to .1 cu. in. Find the volume of each of the following prisms: 6. Base 76.5 sq. ft., altitude 6 ft. 4 in. 7. Base 12 ft. square, altitude 9 ft. 8. The base is a rectangle 8 in. by 9 in., and the altitude is 2 ft. 3 in. Find the volume. 9. Find the altitude of a prism whose volume is 360 cu. ft. and whose base is 80 sq. ft. |