Quest. 47. To divide a cone into three equal parts by sections parallel to the base, and to find the altitudes of the three parts, the height of the whole cone being 20 inches? Ans. the upper part 13.867. the middle part 3.605. the lower part 2:528. QUEST. 48. A gentleman has a bowling green, 300 feet long, and 200 feet broad, which he would raise 1 foot higher, by means of the earth to be dug out of a ditch that goes round it: to what depth must the ditch be dug, supposing its breadth to be every where 8 feet? Ans. 73% feet. Quest. 49. How high above the earth must a person be raised, that he may see of its surface? Ans. to the height of the earth's diameter. QUEST. 50. A cubic foot of brass is to be drawn into wire, of 5 of an inch in diameter; what will the length of the wire be, allowing no loss in the metal ? Ans. 97784:797 yards, or 55 miles 984:797 yards. Quest. 31. Of what diameter must the bore of a cannon be, which is cast for a ball of 24lb. weight, so that the diameter of the bore may be to of an inch more than that of the ball ? Ans. 5.647 inches. QUEST. 52. Supposing the diameter of an iron 9lb. ball to be 4 inches, as it is very nearly; it is required to find the diameters of the several balls weighing 1, 2, 3, 4, 6, 12, 18, 24, 32, 36, and 421b, and the caliber of their guns, allowing of the caliber, or of the ball's diameter, for windage. Answer QUEST. 53. Supposing the windage of all mortars to be zo of the caliber, and the diameter of the hollow part of the shell to be fo of the caliber of the mortar: it is required to determine the diameter and weight of the shell, and the quantity or weight of powder requisite to fill it, for each'of the several sorts of mortars, namely, the 13, 10, 8, 5.8, and 4:6 inch mortar. Answer Calib.of Diameter Wt. of shell Wt. of Wt. of shell of shell. empty powder. filled. mort. 4.6 5.8 8 10 13 4.523 5•703 7.867 9.833 12:783 8.320 16.677 43.764 85.476 187.791 0:583 1.168 3.065 5.986 13:151 8.903 17.845 46.829 91.462 200.942 Quest. 54. If a heavy sphere, whose diameter is 4 inches, be let fall into a conical glass, full of water, whose diameter is 5, and altitude 6 inches; it is required to determine how much water will run over ? Ans. 26•272 cubic inches, or nearly of a pint. QUEST. 55. The dimensions of the sphere and cone being the saine as in the last question, and the cone only full of water; required what part of the axis of the sphere is immersed in the water? Ans. •546 parts of an inch. QUEST. 56. The cone being still the same, and full of water; required the diameter of a sphere which shall be just all covered by the water? Ans. 2.445996 inches. Quest. 57. If a person, with an air balloon, ascend vertically froni London, to such a height that he can just see Oxford appear in the horizon; it is required to determine his height above the earth, supposing its circumference to be 25000 miles, and the distance between London and Oxford 49.5933 miles ? Ans. It's of a mile, or 547 yards 1 foot. QUEST. 58. In a garrison there are three remarkable objects A, B, C, the distances of which from one to another are known to be, AB 213, AC 424, and BC 262 yards; I am desirous of knowing my position and distance at a place or station s, from which I observed the angle asB 13° 30', and the angle cse 29° 50', both by geometry and trigonometry. Answer, QUEST. 59 QUEST. 59. Required the same as in the last question, when the point B is on the other side of ac, supposing AB 9, AC 12, and BC 6 furlongs; also the angle asb 33° 45', and the angle Bsc 22° 30'. Answer, As QUEST. 60. It is required to determine the magnitude of a cube of gold, of the standard fineness, which shall be equal to a sum of 480 million of pounds sterling; supposing a guinea to weigh 5 dwts 94 grains. Ans. 18.691 feet. QUEST. 61. The ditch of a fortification is 1000 feet long, 9 feet deep, 20 feet broad at bottom, and 22 at top; how inuch water will fill the ditch ? Ans. 1158127 gallons nearly, QUEST. 62. If the diameter of the earth be 7930 miles, and that of the moon 2160 miles: required the ratio of their surfaces, and also of their solidities : supposing them both to be globular, as they are very nearly? Ans. the surfaces are as 131 to 1 nearly; and the solidities as 49 to 1 nearly, PRACTICAL EXERCISES CONCERNING SPECIFIC GRAVITY. The Specific Gravities of Bodies are their relative weights contained under the same given magnitude; as a cubic foot, or a cubic inch, &c. The specific gravities of several sorts of matter, are expressed by the numbers annexed to their names in the Table of Specific Gravities, at page 231; from which the numbers are to be taken, when wanted. Note. The several sorts of wood are supposed to be dry. Also, as a cubic foot of water weighs just 1000 ounces avoirdupois, the numbers in the table express, not only the specific gravities of the several bodies, but also the weight of a cubic foot of each in avoirdupois ounces; and hence, by proportion, the weight of any other quantity, or the quantity quantity of any other weight, may be known, as in the fol. lowing problems, PROBLEM I. To find the Magnitude of any Body, from its Weight, As the tabular specific gravity of the body, EXAMPLES. EXAM. 1. Required the content of an irregular block of common stone, which weighs lcwt. or 1121b. Ans. 12285 cubic inches, EXAM. 2. How many cubic inches of gunpowder are there in ilb weight? Ans. 29 cubic inches nearly. ExAM. 3. How many cubic feet are there in a ton weight of dry oak? Ans. 38713 cubic feet, PROBLEM II. To find the Weight of a Body from its Magnitude. EXAMPLES. ExAM, 1. Required the weight of a block of marble, whose length is 63 feet, and breadth and thickness each 12 feet; being the dimensions of one of the stones in the walls of Balbeck ? Ans. 683 15 ton, which is nearly equal to the burden of an East-India ship. EXAM. 2. What is the weight of 1 pint, ale measure, of gunpowder ? Ans. 19 oz. nearly. Exam. 3. What is the weight of a block of dry oak, which measures 10 feet in length, 3 feet broad, and 24 feet deep? Ans. 4335781b. PROBLEM PROBLEM III. To find the Specific Gravity of a Body. CASE 1. When the body is heavier than water, weigh it both in water and out of water, and take the difference, which will be the weight lost in water. Then say, As the weight lost in water, EXAMPLE. A piece of stone weighed 10lb, but in water only 6 lb, required its specific gravity? Ans. 2609, CASE 2. When the body is lighter than water, so that it will not quite sink, affix to it a piece of another body, heavier than water, so that the mass compounded of the two may sink together. Weigh the denser body and the compound mass separately, both in water and out of it; then find how much each loses in water, by subtracting its weight in water from its weight in air; and subtract the less of these re, mainders from the greater. Then say, As the last remainder, EXAMPLE Suppose a piece of elm weighs 15lb in air; and that a piece of copper which weighs 18lb in air, and 161b in water, is affixed to it, and that the compound weighis 6lb in water; required the specific gravity of the elm? Ans. 600. PROBLEM IV. To find the Quantities of Two Ingredients in a Given Compound. Take the three differences of every pair of the three specific gravities, namely, the specific gravities of the compound and each ingredient; and multiply the difference of every two specific gravities by the third. Then say, as the greatest product, is to the whole weight of the compound, so is each of the other products, to the two weights of the ingredients. EXAMPLE |