85. If a piece of silk cost $.80 per yard, at what price shall it be marked, that the merchant may sell it at 10 % less than the marked price, and still make 20% profit? Ans. $1.06. 86. A merchant bought 20 pieces of cloth, each piece containing 25 yd. at $43 per yard on a credit of 9 mo.; he sold the goods at $1 per yard on a credit of 4 mo. What was his net cash gain, money being worth 6 %? 173.85 87. A owes B $1200, to be paid in equal annual payments of $200 each; but not being able to meet these payments at their maturities, and having an estate 10 years in reversion, he arranges with B to wait until he enters upon his estate, when he is to pay B the whole amount, with 8 % compound interest. What sum will B then receive? Ans. $1996.074+. 88. A gentleman who was entitled to a perpetuity of $3000 a year, provided in his will that, after his decease, his oldest son should receive it for 10 yr., then his second son for the next 10 yr., and a literary institution for ever afterward. What was the value of each bequest at the time of his decease, allowing compound interest at 6 %? Ans. To allost son, $22080.28; to second son, $12329.51; to institution, 8/550,23 89. B has 3 teams engaged in transportation; his horse team can perform the trip in 5 days, the mule team in 7 days, and the ox team in 11 days. Provided they start together, and each team rests a day after each trip, how many days will elapse before they all rest the same day? Ans. 24 days. 23 90. A man bought a farm for $4500, and agreed to pay principal and interest in 4 equal annual instalments; how much was the annual payment, interest being 6 %?/298, 67 Ans. $1298.435+. 72.54 91. A bought a piece of property of B, and gave him his bond for $6300, dated Jan. 1, 1860, payable in 6 equal annual instalments of $1050, the first to be paid Jan. 1, 1861. A took up his bond Jan. 1, 1864, semi-annual discount at the rate of 6% per annum on the several payments which fell due after Jan. 1, 1861, being deducted; what sum cancelled the bond? Ans. $2844.20. 92. A gentleman desires to set out a rectangular orchard of 864 trees, so placed that the number of rows shall be to the number of trees in a row, as 3 to 2. If the trees are 7 yards apart, how much ground will the orchard occupy? Ans. 39445 sq. yd. 93. S. C. Wilder bought 25 shares of bank stock at an advance of 6% on the par value of $100. From the time of purchase until the end of 3 yr. 3 mo. he received a semi-annual dividend of 4 %, when he sold the stock at a premium of 11 %. Money being worth 7% compound interest, how much did he gain? Ans. $137.41./3 94. A builder employs a certain number of men, dividing them into companies according to the kind of work they can perform. When he settles with each company at the end of the week, he finds that the number of men in each is such, that they constitute a geometrical 425 progression whose first term is 4, last term 64, and ratio 2. He settles with the men in such a way that, by paying $1.50 per day each to those in the largest company, and $.50 per day to those in the smallest company, and taking one day's wages off one man in each company, he has an arithmetical progression whose common difference is .25. Find the number of companies; the total number of men; the daily wages in each company; and the total sum paid for one week. Ans. 5 companies; 124 men; 1st co., $4 per day; 2d, $6; 3d, $16; 4th, $40; 5th, $96; weekly payment, $972. NOTICE.-A Key to this work is published, containing full and clear solutions to all the examples. MENSURATION. LINES AND SUPERFICIES. 730. The Area of a figure is its superficial contents, or the surface included within any given lines, without regard to thickness. In taking the measure of any line, or surface, we are always governed by some denomination, a unit of which is called the Univ of Measure, (279). PROBLEM I. 731. To find the area of a square or rectangle. RULE. Multiply the length by the breadth. NOTE. For analysis of the principles of this rule see 282. EXAMPLES FOR PRACTICE. 1. How many square yards in the floor, ceiling, and walls of a room 24 feet long, 15 feet wide, and 84 feet high? Ans. 153. 2. The boundary lines of my farm, taken in order, are as follows: The first runs north 38 ch. 20 1.; the second, east 25 ch. 14 1.; the third, south 12 ch. 8 1.; the fourth, west 8 ch. 30 1.; the fifth, south 26 ch. 12 1.; the sixth, west 16 ch. 84 1. to the place of beginning. Required the area. Ans. 74 A. 56.8+ P. 3. If a piece of land 20 rods long contain 240 square rods, what is its width? 4. A surveyor wishes to lay out a rectangular lot of land which shall be 120 rods in length, and contain 70 acres; what must be its width? Ans. 93 rods. 5. A piece of land 8 chains wide contains 40 acres; what is its length in chains? PROBLEM II 732. To find the area of a rhombus or a rhomboid. A Rhombus is a figure having four equal sides and four oblique angles. A Rhomboid is a figure having its opposite sides equal and parallel, and its angles oblique. NOTE. The square, rectangle, rhombus, and rhomboid, having their opposite sides parallel, are called by the general name, parallelogram.. RULE. Multiply the length by the shortest or perpendicular distance between two opposite sides. EXAMPLES FOR PRACTICE. 1. The side of a plat of ground in the form of a rhombus is 36 feet, and the perpendicular distance between either two of the sides is 28 ft. 9 in.; what is the area of the plat? Ans. 3 sq. rd. 218‡ sq. ft. 2. The longer sides of a rhomboid measure 72 rods, and the shorter sides 20 rods; and a perpendicular from the obtuse angle at the extremity of one of the longer sides meets the opposite side 12 rods from the acute angle at its extremity. Required the area of the figure. Ans. 7 A. 32 P. 3. The longer sides of a piece of land in the form of a rhomboid, run north and south, and are 30 rods apart; and the shorter sides are half the length of the longer; if a piece of land in rectangular form was enclosed by lines of the same length as these, respectively, it would contain 14 A. 72 P. What is the area of the rhomboid? Ans. 12 A. 120 P. PROBLEM III. 733. To find the area of a trapezoid. A Trapezoid is a figure having four sides, two of which are parallel. RULE. Multiply one half the sum of the parallel sides by the perpendicular distance between them. EXAMPLES FOR PRACTICE. 1. My farm has four sides, two of which are parallel, and at a distance from each other of 30 ch. 25 1.; the lengths of the parallel sides are 72 ch. 40 1., and 84 ch. 36 1., respectively. How many acres in my farm? Ans. 237 A. 15.92 P. 2. What is the area of a board 12 feet long, 16 inches wide at each end, and 9 in the middle? 3. The boundary lines of a meadow are as follows: the first runs north 36 rods, the second in a northeasterly direction 30 rods, the third south 54 rods, and the fourth west 24 rods, to the place of beginning. Required the area of the meadow. Ans. 6 A. 120 P. PROBLEM IV. 734. To find the area of a triangle. RULE. Multiply one half the base by the altitude, or one half the altitude by the base; or, Multiply the base by the altitude, and divide the product by 2. EXAMPLES FOR PRACTICE. 1. How many square yards in a triangle whose base is 126 feet, and perpendicular 24 feet? Ans. 168 sq. yd. 2. The gable ends of a barn are each 34 ft. wide, and the perpendicular height of the ridge above the eaves is 8 ft.; how many feet of boards will be required to board up both gables? Ans. 272 ft. NOTE 1.-It will readily be seen that the gable may be divided into two rightangled triangles. 3. The area of a right-angled triangle is 48 feet, and the base is 12 feet; what is the perpendicular? NOTE 2. This example is the reverse of the preceding ones. 4. The area of the gable of a certain building is 108 feet, and the perpendicular height of the ridge of the roof above the eaves is 9 feet; what is the width of the building? 5. One side of a triangular field is 18 rods in length, and the perpendicular distance between this side and the opposite angle is 15 rods; what is the area of the field? Ans. 13 A. 2 R. 6. What are the contents of a triangular board, each of whose sides measures 18 inches? Ans. 140.29+ sq. in. PROBLEM V. 735. To find the diameter or the circumference of a circle. It is proved in Geometry that in every circle the ratio between the diameter and the circumference is 3.14159+. Hence RULE. I. To find the diameter; - Multiply the circumference by .31831. II. To find the circumference ; Multiply the diameter by 3.14159. EXAMPLES FOR PRACTICE. 1. What is the circumference of a circle 8 feet in diameter? Ans. 25 ft. 1.59+ in. 2. If the circumference be 49.52 rods, what is the diameter?? Ans. 15.762 rods. 3. What is the length of an arc of 18° in a circle whose radius is 4 ft. 8 in.? Ans. 1 ft. 5.75 in. 4. Within a circular garden 66 chains in circumference, is a circular pond 66 rods in circumference; what is the width of the ring of land surrounding the pond? Ans. 31.51+ rd: 5. The circumference of a cart wheel is 16 ft. 6 in.; what is the diameter? Ans. 5 ft. 3 in. PROBLEM VI. 736. To find the area of a circle. RULE. I. When both diameter and circumference are given; Multiply the diameter by the circumference, and divide the product by 4. II. When the diameter is given;-Multiply the square of the diameter by .7854. III. When the circumference is given ;-Multiply the square of the circumference by .07958. EXAMPLES FOR PRACTICE. 1. The diameter of a circle is 226, and the circumference 710; what is the area? Ans. 40115. 2. What is the area of a circular saw 25 inches in diameter? 3. The circumference of one end of a log is 6 ft. 10 in. ; what is the area? 4. A circular plat contains 6.44598 square chains of land; how many rods in the circumference? Ans. 36. 5. What is the area of a quadrant whose radius is 4 rode? Ans. 12.5664 sq. rd. 737. A Solid or Body is a magnitude which has length, breadth, and thickness. In estimating the solid contents of a body, we are always governed by some denomination, a unit of which is called the Unit of Measure, (287). 738. A Cylinder is a body whose bases or ends are equal and parallel circles, and whose side is a uniform curved surface |