and the vol. of the lower portion of the stack 4.π Also, the volume of the upper, or conical, portion of II. Let the base and section be oblong, instead of circular, and the upper part still terminate in a point; then we have a pyramidal, instead of a conical, upper portion; and its volume height x area of section at its base. 3 Also, the lower portion is a frustum of a pyramid, instead of a cone; and its volume =ht. x area of mid-section, ht.x sum of areas of its highest and lowest sections. III. If the base be as in the last case, and the upper portion do not terminate in a point, but in a ridge parallel and equal to the length of the stack, then the upper part will be a triangular prism, of which ABE is the base, and its length the length of the ridge. case. Hence, volume of prism = length× area of ABE, = length of ridge ×AOxBO. The lower part is a pyramidal frustum, as in the last Ex. 1. The heights of the upper and lower portions of a round stack are 4 ft. and 7 ft. respectively; and the girths of the highest and lowest parts of the frustum are 30 ft. and 24 ft.; find the volume of the stack. The circumference of the mid-section of the frus30+24 tum = =27; 2 =108 cub. ft., nearly; .. the whole volume=514 cub. ft., nearly, Ex. 2. A stack stands on a rectangular base 13 ft. in length, and 9 ft. in width; the horizontal section at the eaves is 16ft. in length, and 11 ft. in width; and the height of this portion is 7 ft. The upper portion terminates in a point, and its height is 44 ft. Find the volume of the stack. Area of highest section of the frustum=16×11=176 sq. ft. lowest .=13× 9=117 Area of mid-section=144×10=145 sq. ft. ; .. vol. of lower portion=7×145=1015 cub. ft. Also vol. of upper pyramidal portion 4×16×11 3 =264 cub. ft. .. whole volume =1279 cub. ft. =142) cub. yds. Ex. 3. The same data as in Ex. 2, except that the upper portion, instead of tapering to a point, terminates in a ridge of the same length as the horizontal section which forms its base. Vol. of lower portion (as in Ex. 2)=1015 cub. ft. 16×11×4 cub.ft. 2 =396 cub. ft.; .. whole volume =1411 cub. ft.= 1563 cub. yds. NOTE. A cubic foot of old hay will weigh about 8 lbs., on the average, as proved by experiment. Hence the weight of a stack will readily be found, when its volume, or content, has been determined. Two cwt. per cubic yard will not be far wrong. EXERCISES N. (1) Find the cost of a block of stone in the form of a parallelopiped, whose edges are 3, 5, and 8, feet, at 2s. 2d. per cubic foot. (2) What length must I cut off from a broad, and 1 ft. thick, for the sum of £2. 5s., of 10d. per cubic foot? Ans. £13. plank 2 ft. at the rate Ans. 18 ft. (3) What would the painting, of the whole piece cut off in the last Ex., cost, at 1d. per square foot? Ans. 11s. (4) The bottom of a cistern contains 7 sq. ft. 101 sq. in.; how deep must it be to hold 82 gallons, if 277 cub. in. make 1 gallon? Ans. 1 ft. 8 in. (5) A right-angled triangle, whose sides are 3, 4, and 5, inches, is made to turn round upon the side whose length is 4 in., thus describing a right cone; find the surface and volume of the cone. (1) Ans. 47 sq. ft. (2) Ans. 37 cub. ft. (6) A rectangular parallelogram, 7 inches long, and 1 inch broad, is turned round about one of its longer sides, and describes a cylinder; find the surface and volume of the cylinder. (1) Ans. 44 sq. ft. (2) Ans. 22 cub. ft. (7) A cylindrical shaft, 105 yds. deep, and 2 yds. wide, was to be excavated, at the rate of £1. per yard in depth; but the rate is afterwards changed to one of 6s. 8d. per cubic yard excavated; what difference is there in the cost? Ans. It costs £5. more. (8) A right prism, whose ends are equilateral triangles, having their sides each 3 in., is 16 in. long; find its surface and volume. (1) Ans. 1 sq. ft. 24 sq. in. (2) Ans. 84-868 cub. in. (9) An oblique prism has a polygonal base of the form described in (226), where the diagonal AD=4.5 in., and AC= 4.8 in.; also the perpendiculars Bb, Dd, Ee, are 1.2, 2.5, and 16, inches, respectively; and the perpendicular height of the prism is 10 in.; find its volume. Ans. 116 cub. in. (10) A pyramid of marble has for its base a regular hexagon, whose side is 1 ft.; and the height of the pyramid is 9 ft.; what is the cost, at 10s. per cubic foot? Ans. £3. 17s. 11.28d. (11) Some blocks of wood, 1 foot high, and having their ends 4 inches square, are cut into hexagonal prisms, with as little waste as possible; find the cost per 1000, at the rate of 2s. 6d. per cubic foot of manufactured material. Ans. £10. 8s. 4d. (12) An hour-glass is made of two equal cones joined at their vertices; the vertical angle is 60o, and the depth of the sand when level in one of the cones is 3 inches; find the volume of sand which must pass into the lower cone per minute, so that the upper cone may be emptied in 1 hour. 11 Ans, cub. in. 70 (13) Find the cost of lining a cylindrical shaft, 30 yards deep, and 1 yards broad, with wood 3 inches. thick, supposing the cost of material and labour to be at the rate of 1s. 9d. per cubic foot. Ans. £10. 6s. 3d, (14) A cubical mass of metal, whose edge is 3.35 inches, is drawn out into a cylindrical wire 67 inches long; find the area of a section of it perpendicular to its length. Ans. 561125 sq. in. (15) The adjacent edges of a rectangular box are 3.428571, 5·142857, and 10-285714, inches; find the cost of gilding its exterior at 13d. per square inch. Ans. £1. 10s. 104d. (16) A solid spherical ball of copper is hammered into a circular plate of one inch uniform thickness. Find the diameter of the plate. Ans. 2.828 feet. (17) How many bullets of a quarter of an inch in diameter can be cast from the metal of a spherical ball 3 inches in diameter, supposing no waste in the process? Ans. 1728. (18) A river with an average depth of 30 feet, and 200 yards wide, is flowing at the average rate of 4 miles an hour; find how many cubic feet of water run into the sea per minute; also the number of tons, supposing a cubic foot of water to weigh 1000 ounces. (1) Ans. 6,336,000 cub. ft. (2) Ans. 176785 tons. (19) What is the number of cubic feet in the volume of an hexagonal room, each side of which is 20 ft. long, and the walls 30 ft. high, and which is finished above with a roof in the form of an hexagonal pyramid 15 ft. high? Ans. 155880 cub. ft. (20) Find the cost of painting the walls and ceiling of the room, described in the last Ex., at 1s. per sq. yd. Ans. £26. (21) What is the solid content of a sphere, whose lineal circumference is 6 yds.? Ans. 4 cub. yds. 54 ft. (22) What is the solid content of a sphere, when its surface is equal to that of a circle 8 ft. in diameter ? Ans. 33 cub. yds. 144 ft. (23) Required the cost of a globe of 25 in. diameter, which is to be paid for at 6d. the square inch on the surAns. £49. 2s. 14d. face. |