2. A man purchased 12 pears: he was to pay 1 farthing for the 1st, 2 farthings for the 2d, 4 for the 3d, and so on doubling each time: what did he pay for the last? 3. A gentleman dying left nine sons, and bequeathed his estate in the following manner: to his executors £50; his youngest son to have twice as much as the executors, and each son to have double the amount of the son next younger: what was the eldest son's portion? 4. A man bought 12 yards of cloth, giving 3 cents for the 1st yard, 6 for the 2d, 12 for the 3d, &c.: what did he pay for the last yard? CASE II. 323. Having given the ratio and the two extremes to find the sum of the series. Subtract the less extreme from the greater, divide the remainder by 1 less than the ratio, and to the quotient add the greater extreme: the sum will be the sum of the series. EXAMPLES. 1. The first term is 3, the ratio 2, and last term 192; what is the sum of the series? 192 3189 difference of the extremes, 2 − 1 = 1) 189 (189; then 189 + 192 381 Ans. 2. A gentleman married his daughter on New Year's day, and gave her husband 1s. towards her portion, and was to double it on the first day of every month during the year: what was her portion? 3. A man bought 10 bushels of wheat on the condition that he should pay 1 cent for the 1st bushel, 3 for the 2d, 9 for the 3d, and so on to the last: what did he pay for the last bushel and for the 10 bushels? 4. A man has six children; to the 1st he gives $150, to the 2d $300, to the 3d $600, and so on, to each twice as much as the last: how much did the eldest receive, and what was the amount received by them all? QUEST.-223. How do you find the sum of the series? MENSURATION. 324. Mensuration is the process of determining the contents of geometrical figures, and is divided into two parts, the mensuration of surfaces and the mensuration of solids. 1 Foot. 325. Surfaces have length and breadth. They are measured by means of a square, which is called the unit of surface. A square is the space included between four equal lines, drawn perpendicular to each other. Each line is called a side of the square. If each side be one foot, the figure is called a square foot. 1 Foot. Square foot. If the sides of a square be each four feet, the square will contain sixteen square feet. For, in the large square there are sixteen small squares, the sides of which are each one foot. Therefore, the square whose side is four feet, contains sixteen square feet. The number of small squares that is contained in any large square is always equal to the product of two of the sides of the large square. As in the figure, 4×4=16 square feet. The number of square inches contained in a square foot is equal to 12×12=144. 326. A triangle is a figure bounded by three straight lines. Thus, BAC is a triangle. QUEST.-324. What is mensuration? 325. What is a surface? What is a square? What is the number of small squares contained in a large square equal to 326. What is a triangle ? The three lines BA, AC, BC, are call ed sides: and the three corners, B, A, and C, are called angles. The side AB is called the base. When a line like CD is drawn making A B the angle CDA equal to the angle CDB, then CD is said to be perpendicular to AB, and CD is called the altitude of the triangle. Each triangle CAD or CDB is called a right-angled triangle. The side BC, or the side AC, opposite the right angle, is called the hypothenuse. The area or contents of a triangle is equal to half the product of its base by its altitude (Bk. IV. Prop. VI).* EXAMPLES. * 1. The base, AB, of a triangle is 50 yards, and the perpendicular, CD, 30 yards: what is the area? 2. In a triangular field the base is 60 chains, and the perpendicular 12 chains: how much does it contain ? 3. There is a triangular field, of which the base is 45 rods and the perpendicular 38 rods: what are its contents ? 4. What are the contents of a triangle whose base is 75 chains and perpendicular 36 chains? 327. A rectangle is a four-sided figure like a square, in which the sides are perpendicular to each other, but the adjacent sides are not equal. All the references are to Davies' Legendre. QUEST.-326. What is the base of a triangle? What the altitude? What is a right-angled triangle? Which side is the hypothenuse ! What is the area of a triangle equal to 327. What is a rectangle? 328. A parallelogram is a four-sided figure which has its opposite sides equal and parallel, but its angles not rightangles. The line DE, perpendicular to the base, is called the altitude. D E 329. To find the area of a square, rectangle, or parallelo gram, Multiply the base by the perpendicular height, and the product will be the area (Bk. IV. Prop. V). EXAMPLES. 1. What is the area of a square field of which the sides are each 66.16 chains? 2. What is the area of a square piece of land of which the sides are 54 chains? 3. What is the area of a square piece of land of which the sides are 75 rods each? 4. What are the contents of a rectangular field, the length of which is 80 rods and the breadth 40 rods ? 5. What are the contents of a field 80 rods square? 6. What are the contents of a rectangular field 30 chains long and 5 chains broad? 7. What are the contents of a field 54 chains long and 18 rods broad? 8. The base of a parallelogram is 542 yards, and the perpendicular height 720 feet: what is the area? 330. A trapezoid is a four-sided figure ABCD, having two of its sides, AB, DC, parallel. The perpendicular EF is called the altitude. DE C A F B QUEST.-328. What is a parallelogram? 329. How do you find the area of a square, rectangle, or parallelogram? 330. What is a trapezoid ? 331. To find the area of a trapezoid, Multiply the sum of the two parallel sides by the altitude, and divide the product by 2, and the quotient will be the area (Bk. IV. Prop. VII). EXAMPLES. 1. Required the area or contents of the trapezoid ABCD, having given AB=643.02 feet, DC=428.48 feet, and EF =342.32 feet. We first find the sum of the sides, and then multiply it by the perpendicular height, after which, we divide the product by 2, for the area. OPERATION. 643.02 +428.48: 1071.50 = 2 2. What is the area of a trapezoid, the parallel sides of which are 24.82 and 16.44 chains, and the perpendicular distance between them 10.30 chains? 3. Required the area of a trapezoid whose parallel sides are 51 feet, and 37 feet 6 inches, and the perpendicular distance between them 20 feet 10 inches. 4. Required the area of a trapezoid whose parallel sides are 41 and 24.5, and the perpendicular distance between them 21.5 yards. 5. What is the area of a trapezoid whose parallel sides are 15 chains, and 24.5 chains, and the perpendicular height 30.8 chains? 6. What are the contents when the parallel sides are 40 and 64 chains, and the perpendicular distance between them 52 chains ? QUEST.-831. How do you find the area of a trapezoid? |