Denominate Numbers-Lumber. PROBLEMS. NOTE 4.-A 14-inch board contains more lumber than a l-inch board of the same width and length. A 14-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, 14-in. boards, each of which is 12 inches wide ? 4. How much lumber in 4, 16-ft., 14-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. 6. 32 pieces 6 x 6, 12. 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. Denominate Numbers-Lumber. PROBLEMS. Note 6.-In the measurement of timbers of all sizes it is custom. ary 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 94 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.* 8. 6 pieces 12 x 12, 24. 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 : $16 per 1. 26, 16-foot, 6-in. fence boards @ $15 per M. M. $16 M. $15 per M. $15 M. * Find the number of feet in 1 piece of 2 X 4, 14 (nearest whole number of feet) and multiply by 16. 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 43 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. 346. MISCELLANEOUS PROBLEMS. 1. Find the solid content of an octagonal right prism the area of whose base is 24 square inches and whose altitude is 15 inches. 2. What is the solid content of a cylinder or of any right prism, the area of whose base is 30 square inches and whose altitude is 12 inches ? 3. How many cubic feet of earth must be removed to dig a well 6 feet in diameter and 20 feet deep ? * 4. Find the approximate number of feet of 14-in. lumber required to inake the lining of the sides of a cylindrical silo that is 20 feet in diameter and 30 feet deep. 5. Find the approximate number of cords of rough stone in a cylindrical pile that is 16 feet in diameter and six feet deep. 6. Find the approximate number of brick necessary for a solid cylindrical foundation that is 9 feet in diameter and 4 feet high. 7. If the average specific gravity of the brick and mortar used in the foundation described in problem 6, is 1.9, how much does the entire foundation weigh? 8. Find the weight in kilograms of a column of water 1 decimeter square and 10 meters deep. 9. Find the weight in pounds of 1000 feet of white pine 1inch boards, the specific gravity being .6. 10. Find the weight of a load (1 cubic yard) of wet sand, the specific gravity being exactly 2. * The exact number of cubic feet cannot be expressed iis figures. An approxi. mation that will arswer many practical purposes may beobtained by regarding the circle (base) as of its circumscribed square. If an answer more nearly accurate is required use .78 instead of 1. DENOMINATE NUMBERS. Capacity. 347. The standard unit of capacity used in measuring liquids is a gallon. A gallon equals 231 cubic inches. Liquid Measure. Observe that 1 cubic foot nearly 71 gallons. A kerosene barrel contains about 52 gallons. It equals nearly 7 cubic feet. PROBLEMS. 1. Find the capacity (approximate or exact), in gallons, of a rectangular tank 3 ft. by 4 ft. by 8 ft. 2. Find the approximate capacity, in gallons, of a cylindrical tank 4 feet in diameter and 4 feet deep. 3. Find the approximate capacity, in barrels (31} gal.), of a rectangular tank 2 ft. by 4 ft. by 12 ft. 4. Find the approximate capacity, in barrels (31) gal.), of a cylindrical cistern 6 ft. in diameter and 6 ft. deep. 5. Find the approximate capacity, in barrels (31 gal.), of a cylindrical cistern 12 ft. in diameter and 6 ft. deep. 6. Find the approximate capacity, in barrels (31+ gal.), of a cylindrical cistern 12 ft. in diameter and 12 ft. deep. |