MISCELLANEOUS EXAMPLES. 1. A merchant engages a clerk at the rate of $ 20 for the first month, $25 for the second, $ 30 for the third, &c., thus increasing his salary by $ 5 per month. How long must the clerk retain his situation, so as to receive on the whole as much as he would have received had his salary been fixed at $ 52.50 per month ? Ans. 14 months. 2. A mason has plastered 3 rooms; the ceiling of each is 20 feet by 16 feet 6 inches, the walls of each are 9 feet 6 inches high, and 90 yards are to be deducted for doors, windows, &c. For how many yards must he be paid ? Ans. 25lyd. 1ft. 6'. 3. A man of wealth, dying, left his property to his ten sons, and the executor of his will, as follows: to his executor, $ 1024; to his youngest son, as much and half as much more; and increasing the share of each next elder in the ratio of 13. What was the share of the eldest? 4. A butcher, wishing to buy some sheep, asked the owner how much he must give him for 20; on hearing his price, he said it was too much ; the owner replied, that he should have 10, provided he would give him a cent for each different choice of 10 in 20, to which he agreed. How much did he pay for the 10 sheep, according to the bargain? Ans. $ 1847.56. 5. If 340 square feet of carpeting are required to cover the floor of a room, how many yards will be required, provided the width of the carpeting is 3 feet 9 inches ? Ans. 30yd. Sin. 6. If a clergyman's salary of $ 700 per annum is 6 years in arrears, how much is due him, allowing compound interest at 6 Ans. $ 4882.72. 7. Suppose a clock to have an hour-hand, a minute-hand, and a second-hand, all turning on the same center. At 12 o'clock all the hands are together and point at 12. (1.) How long will it be before the second-hand will be between the other two hands, and at equal distances from each? Ans. 60723 seconds. (2.) Also before the minute-hand will be equally distant between the other two hands? Ans. 61654 seconds. (3.) Also before the hour-hand will be equally distant between the other two hands? Ans. 594 seconds. per cent. ? MENSURATION. DEFINITIONS. A B D 602. A point is that which has neither length, breadth, nor thickness, but position only. 603. A line is length, without breadth or thickness. A straight line is one which has the same direction in its whole extent; as the line A B. A curved line is one which continually changes its direction; as the line C D. 604. An angle is the inclination or opening of two lines, which meet in a point. A right angle is an angle formed by a straight line and a perpendicular to it; as the angle A B C. An acute angle is one less than a right angle; as the angle E BC. An obtuse angle is one greater than a right angle; } as the angle FBC. A с B E B B 605. A surface is that which has length and breadth, without thickness. A plane surface, or simply a plane, is that in which, if any two points whatever be taken, the straight line that joins them will lie wholly in it. Every surface, which is not a plane, or composed of planes, is a curved surface. 606. The area of a figure is its quantity of surface; and is estimated in the square of some unit of measure, as a square inch, a square foot, &c. 607. A solid, or body, is that which has length, breadth, and thickness. 608. The solidity, or volume of a solid, is estimated in the cube of some unit of measure; as a cubic inch, a cubic foot, &c. 609. MENSURATION is the process of determining the areas of surfaces, and the solidity or volume of solids. MENSURATION OF SURFACES. 610. A plane figure is an enclosed plane surface ; if bounded by straight lines only, it is called a rectilineal figure, or polygon. 'The perimeter of a figure is its boundary, or contour. 611. Three-sided polygons are called triangles; those of four sides, quadrilaierals ; those of five sides, pentagons, and so on. TRIANGLES. 612. An equilateral triangle is one whose sides are all equal; as CAD. NOTE. — The line A B, drawn from the angle A perpendicular to the base C D, is the altitude of the triangle CAD. А с D B An isosceles triangle is one which has two of its sides equal; as EFG. A scalene triangle is one which has its three sides unequal; as HIJ. H M A right-angled triangle is one which has a right angle ; as KL M. area. 613. To find the area of a triangle. Multiply the base by half the altitude, and the product will be the Or, Add the three sides together, take half that sum, and from this subtract each side separately; then multiply the half of the sum and these remainders together, and the square root of this product will be the area. Ex. 1. What are the contents of a triangle whose perpendicular height is 12 feet, and whose base is 18 feet ? Ans. 108 feet. 2. There is a triangle, the longest side of which is 15.6 feet, the shortest side 9.2 feet, and the other side 10.4 feet. What are the contents ? Ans. 46.139+ feet. 3. The triangular gable of a certain building has a base of 40 feet and an altitude of 15 feet; how many square feet of boards will cover it? Ans. 300 sq. ft. 4. The perimeter of a certain field in the form of an equilateral triangle is 336 rods; what is the area of the field ? Ans. 33 acres 152 sq. rd. QUADRILATERALS. 614. A parallelogram is any quadrilateral whose opposite sides are parallel. C 615. A rectangle is any right-angled parallelogram ; as A B CV. D A B 616. A square is a parallelogram whose sides are equal, and whose angles are right angles; as E F G H. 617. A rhombnis is a parallelogram whose sides are equal, and whose angles are not right angles; as IJKL. I P 618. A rhomboid is a parallelogram whose angles are not right angles; as MNÓP. M N NOTE. — The altitude of a parallelogram is the perpendicular distance between any two of its parallel sides taken as bases, as the line P Q, drawn between two sides of the rhomboid MNOP, and perpendicular to the sides MN and OP. T 619. A trapezoid is a quadrilateral which has only two of its sides parallel; as RSTU. U W X 620. A trapezium is a quadrilateral which has no two sides parallel ; as WXYZ. Note. A diagonal of a quadrilateral, or of any polygon of more than four sides, is a straight line which joins the vertices of two opposite angles, or of two angles not adjacent; as the line X Z joining vertices of opposite angles of the trapezium WXYZ. 621. To find the area of a parallelogram. Multiply the base by the altitude, and the product will be the area. Ex. 1. What are the contents of a board 15 feet long and 2 feet wide ? Ans. 30 feet. 2. A rectangular state is 128 miles long and 48 miles wide. How many square miles does it contain ? Ans. 6144 miles. 3. The base of a rhomboid being 12 feet, and its height 8 feet, required the area. Ans. 96 feet. 4. Required the area of a rhombus of which one of the equal sides is 358 feet, and the perpendicular distance between it and the opposite side is 194 feet. Ans. 69452 sq. ft. 5. The largest of the Egyptian pyramids is square at its base, and measures 693 feet on a side. How much ground does it cover ? Ans. 11 acres 4 poles. 6. What is the difference between the area of a floor 40 feet square, and that of two others, each 20 feet square ? Ans. 800 feet. 7. There is a square whose area is 3600 yards; what is the side {43.185 yards, breadth of the of a square, and the breadth of a walk along each side and each end of the square, which shall take up just one half of the whole ? Ans. 8.78+ yards, breadth of the walk. 622. To find the area of a trapezoid. Multiply half of the sum of the parallel sides by the altitude, and the product is the area. Ex. 1. If the parallel sides of a trapezoid are 75 and 33 feet, and the perpendicular breadth 20 feet, what is the area ? Ans. 1080 sq. ft. 2. Required the area of a meadow in the form of a trapezoid, whose parallel sides are 786 and 473 links, and whose altitude is 986 links. Ans. 6 acres 33 rods 3 yards. 623. To find the area of a trapezium. Divide the trapezium into two triangles by a diagonal, and then find the areas of these triangles; thei' sum will be the area of the trapezium. Ex. 1. Required the area of a garden in the form of a trapezium, of which the four sides are 328, 456, 572, and 298 feet, and the diagonal, drawn from the angle between the first and second sides, 593 feet. Ans. 3 acres 1 rood 31 rods 29 yards 3.85 feet. 2. Given one of the diagonals of a field, in the form of a trapezium, equal 17 chains 56 links, to compute the area, the perpendiculars to that diagonal from the opposite angles being 8 chains 82 links, and 7 chains 73 links. Ans. 14 acres 2 roods 5 rods. PENTAGONS, HEXAGONS, &c. 624. A pentagon is a polygon of five sides ; a hexagon, one of six sides; a heptagon, one of seven sides; an octagon, one of eight sides; a nonagon, one of nine sides; and so on for a decagon, undecagon, dodecagon, &c. D 625. A regular polygon is one whose sides and angles are equal; as the pentagon Á B C D E. A B 626. To find the area of a regular polygon. Multiply the perimeter by half the perpendicular let fall from the centre upon one of the sides. Or, Multiply the square of one of the sides by the number against the polygon in the following TABLE. Nonagon, 6.181824 Decagon, 7.694209 Undecagon, 9.365641 Dodecagon, 11.196152 |