rectangle are divided. Thus, if the sides of a rectangle be 3 feet and 4 feet, the rectangle will contain 12 square feet; where I square foot is the unit of measurement. In like manner, if we consider a rectangular block to be divided into slices of equal thickness by planes parallel to the faces of the solid, it is plain that the block will be divided into a number of equal cubes, each of which may be taken as the unit of measurement; and the number of cubes in any one slice will be equal to the product of the numbers representing the number of units in its length and breadth; and as each slice contains the same number of cubes, the whole content will be equal to the number of cubes in one slice multiplied by the number of slices, that is, by the number of units in the thickness of the block. This gives the ordinary rule:"Multiply the length, breadth, and thickness together, and the result is the solidity." Ex. I. To find the number of square feet in a rectangle whose length is 7 feet 5 inches, and breadth 5 feet 4 inches. 7 ft. 5 in. 89 inches 5 ft. 4 in. = = = 64 356 534 144) 5696 (39 432 1376 1296 80 Ans. 39 sq. ft. 80 sq. inches. The latter method is sometimes advantageous in practice for since shillings and pence are duodecimals, if we wish to find the value of the above result at 2s. 7d. per square foot; we proceed as follows: Ex. 2. To find the cost of flagging a floor 20 ft. 3 in. long, and 7 ft. 9 in. broad, at 35. 9d. per square yard. EXAMPLES. [The following dimensions are all expressed in duodecimals]. I. Find the value of 24 ft. 5 in. at 7d. per foot. Ans. £8. 10s. Iid. 2. Find the value of 36.4.6 feet at 3s. 3. 4. 71d. per foot. Ans. 6. 115. Id. Find the value of 46. 10. 9 yards at 5s. 43d. per yard. Find the price of a board 22 ft. 7 in. long, and 151 inches broad, at 24d. per square foot. Ans. 6s. old. 5. Find the cost of plastering a wall 65 ft. 9 in. long, and 9 ft. 6 in. high, at 131d. per square yard. Ans. £3. 18s. old. 6. Find the value of a rectangular block of stone whose dimensions are 8 ft. 3 in., 6 ft. 8 in., 5 ft. 4 in., at 5s. 4d. per foot. Ans. £78. 4s. 5d. RIGHT-ANGLED TRIANGLE. Let ABC be a right-angled triangle, having the right angle ABC. The square described on AC is equal to the squares described on AB, BC. I. Given the base and perpendicular to find the hypothenuse. Square the sides, add them together, and extract the square root. Ex. I. The base of a right-angled triangle is 85, and the perpendicular 132; find the hypothenuse. 852= 7225 1322=17424 24649 (157 Ans. Ex. 2. I 25) 146 125 307) 2149 2149 Find the hypothenuse of a right-angled triangle whose base is 67 yards, and perpendicular 51 yards. N. B. The last two figures in the square root are obtained by contracted division. Ex. 3. Find the hypothenuse of a right-angled triangle whose base is 43ft. 4 in., and perpendicular 19 ft. 3 in. I. The base of a right-angled triangle is 88, and the perpendicular 105; find the hypothenuse. Ans. 137. 2. The base of a right-angled triangle is 264, and the perpendicular 23; find the hypothenuse. Ans. 265. 3. Find the hypothenuse of a right-angled triangle whose base is 44, and perpendicular 117. Ans. 125. 4. Find the hypothenuse of a right-angled triangle whose base is 5 ft. 9 in., and perpendicular 21 ft. 8 in. Ans. 22 ft. 5 in. |