MISCELLANEOUS QUESTIONS. 73.-What would a circular reservoir, whose diameter at top is 40 yards, at bottom 383 yards, and its side or slant depth 11 feet, cost lining with brick-work, at 3s. 10d. the square yard? 73.—The four sides of a field, whose diagonals are equal to each other, are 25, 35, 31, and 19 poles respectively: what is the area? 74.-What is the length of a cord that will cut off one-third of the area, from a circle whose diameter is 289. 75.-A cable which is 3 fet long, and 9 inches in compass, weighs 22lbs.; what will a fathom of that cable weigh whose diameter is 9 inches? 76.-Required that number, of which p times the mth power is equal to g times the (m+2)th power? 77.-Given x-Vy=3 and /x+y=7, required x and y. 78.-The convex superfices of the common parabolic conoid will be least when its abscissa is 0.903084 × V(s); and its greatest ordinate 0.839605 × √(s), (s being equal its solidity.) Query, the demonstration? 79.—In a right-angled triangle, there are given the ratio of the sides as 3 to 4, and the difference between the area of its inscribed circle and inscribed square=20.2825. Required the area and sides of the triangle. 80.-There is an isosceles triangle, whose side, being added to the square root of three times itself, is 60 inches. Query, the side and base of the triangle, whose area is a maximum? 81.—It was my chance to be surveying a piece of land, in the form of a rectangular triangle, whose sides I measured, and found the hypothenuse Just 30 chains; but, night approaching, I found I had blunderingly taken the other two sides both in one sum, just 42 chains 36 links; yet I hope some ingenious gentleman will, from these data, find me the sides and argles by a simple equation. 82.-To find that fraction which exceeds its cube by the greatest quantity possible. 83.-To determine the greatest rectangle that can be inscribed in a given triangle. 84.-Required the fluxion of (ax3 + b)* + 2√/(a* — x°) × (x − 1). 85.-Required the formula for summing an infinite series of fractions, whose numerations are in arithmetical, and denominations in geometrical, progression. 86.-There is a triangular field, whose content is known to be = 15 acres, 2 roods, and 16 perches; the perimeter 78 chains; and one of the angles 126° 52′ 12′′. It is required to find the sides of the field separately, by a general theorem, that may be of use to the practical surveyor. 87.-There is a triangular garden, the lengths of whose sides are 200, 198, and 178 yards. Now, there is a dial so placed in the garden, that, if walks be made from each of the three angles to the dial, they will exactly divide the said garden into three equal parts: from whence is required the length of each walk. 88.--Given the two stationary distances AB=60, B and BC= 80 miles, making a right angle at B; and supposing a messenger sets out from A, in order to travel from thence to C, in the shortest time possible, but, through the unevenness of the road, and other mpediments, he can travel no more than three miles per hour, until he arrives at D Now, supposing the angle BAD to be 35°, in what time will he arrive at C? 89. Given the rectangle of two sides of a plane triangle = 16800, and its perimeter = 560. Query, its area when a maximum? 90.-In a plane triangle ABC, there is given the sides AC and BC equal to 24 and 30 poles respectively; and supposing a circle inscribed in the same, so as to touch all its sides, a line drawn from the angle C to the centre thereof is found to measure 12 poles. Query, the base of the triangle by a simple equation. 91. Supposing the ball at the top of St. Paul's church to be 6 feet in diameter, what would the gilding of it come to, at 34d. per square incl. ! 92.-A person wants a cylindrical vessel of 3 feet, that shall hold twice as much as another of 28 inches deep, and 46 inches in diameter: what must be the diameter of the required vessel? 93.-Two porters agreed to drink off a quart of strong beer between them, at two pulis, or a draught each: now, the first having given it black-eye, as it is called, or drunk till the surface of the liquor touched the opposite edge of the bottom, gave the remaining part of it to the other: what was the difference of their shares, supposing the pot to be the frustum of a cone, the depth of which is 5.7 inches, the diameter at the top 3.7 inches, and that of the bottom 4.23 inches? 95.-Required the fiuxion of SV(1+x®)+xĮ §. 96.-Required the fluxion of the arc a whose tangent= 97.-The three sides of a triangle being given, to find the segments formed by letting fall a perpendicular from the vertical angle upon the base, the perpendicular itself, the area of the triangle, and the radii of inscribed and circumscribed circles? 98. In any right-angled triangle, the area (= 294), and the difference between the hypothenuse and perpendicular (= 14), being given; to find all the other parts of the triangle by a simple equation? 99.-There is a triangular prism, whose angles are 50, 60, and 70°; also its whole superfices is 100 feet. The solidity (being a maximum,) is required. 100. I happen'd, one day, at a tavern to be, Carousing and drinking of port mighty free, "Pray desist," says my friend, " your skill quickly try; I being fond of the office, we both did repair To this three-corner'd ; but, when we came there, We found that with water 'twas so overflown. Not a side, nor an angle, could then have been known; At which I was puzzled. Not knowing what to do, To assist me, my friend then told me he knew That there were three straight paths from the angles proceeded, Namely) making right angles: and their lengths, he rememb'red, And no more is there given to find the content; As the area and sides of this meadow to find. 101.-A and B carried 100 eggs, between them, to market, and each received the same sum. If A had carried as many as B, he would have received 18 pence for them; and, if B had taken only as many as A, he would have received only 8 pence. How many had each? 102.-Find two numbers, such that the square of the greater, multiplied by the less, may be 448, and the square of the less, multiplied by the greater, may be 392. 103.-How many 3-inch cubes can be cut out of a 12-inch cube? 104.-A farmer borrowed part of a hay-rick of his neighbour, which measured 6 feet every way, and paid him back again by two equal cu bical pieces, each of whose sides were 3 feet. Query, whether the lender was fully paid? 105.-What will the painting a conical church-spire come to, at 3d. per yard, supposing the circumference of the base to be 64 feet, and the altitude 118 feet? 106.-Required the greatest right-angled triangle that can be inscribed in the quadrant of a given circle, supposing the hypothenuse of the triangle to be a radius of the circle. 109.-Required a vulgar fraction equivalent to the circulating decimal fraction 0.18 18 18, &c. 110.-The earth having been re-peopled, after the flood, by the three sons of Noah and their three wives, in what ratio must the population have increased each year, so that there may have been a million inhabitants at the end of 200 years? 111.-A joist is 84 inches deep, and 34 broad; what will be the dimensions of a scantling just as big again as the joist, that is, 44 inches broad? 112.-A roof, which is 24 feet 8 inches by 14 feet 6 inches, is to be covered with lead, at 8 lbs. to the foot: what will it come to, at 18s. per cwt? 113. If the side of an equilateral triangle be 10 chains, what will be the side of another equilateral triangle, whose area is 4th of the former? 114. Given the sum of a side, and its adjacent segment =90, and the ratio of the said segment to the base as 2:5. Required the dimensions of the triangle when the area is a maximum, 115.-There is a triangle ABC inscribed in a circular quadrant, whose F longest side AB is a chord within it, and is 9 chains per Gunter. In this triangle the perpendicular BD is drawn, and there is given AB = BD+ CD, to find the radius of the quadrant. 116. A gentleman has a triangular piece of land, whose sides are in the proportion of 3, 4, and 5, and the area of it is equal to the cube of one-fifth part of the base; required the sides and area in numbers. 117. Given the difference of the segments of the base made by a line bisecting the vertical angle, the difference of the sides, and the difference of their squares, to construct the triangle. 118.-From a given circle to cut off an arc, such that the rectangle under its versed sine, and the cosine of its double, may be a maximum. 119.—Given x3 — y3 — 56, and x − y = 16 . xy ; required x and y. 120.-Required the fluxion of u = (cos. x) яu. x ̧ 121-Required the fluxion of an arc x, of which the sine 2a √(1-12). 122.-Given x2 - ; required x and y. 123. Suppose the expence of paving a semicircular plot at 28. 4d. per foot, amounted to 10.; what is the diameter of it? 124 --Seven men bought a grinding-stone of 60 inches in diameter, each paying one-seventh part of the expence: what part of the diameter must each grind down for his share? 125.-A garden, 100 feet long and 80 feet broad, is to have a gravelwalk, of an equal width, half round it: what must the width of the walk be, so as to take up just half the ground? 126.-How many gallons, wine-measure, will a cistern hold, supposing its length and breadth at top to be 5 and 4 feet respectively, and at bottom 4 and 3 feet; the perpendicular depth being 34 feet? 127. A malster has a kiln, that is 16 feet 6 inches square, which he wants to pull down, and to build a new one, that will dry three times as much at a time as the old one: what must be the length of its side? 128.-Suppose that the population of the United States increases every year by a hundredth-part, in how many years will the number of inha bitants be ten times what it now is? 129.--In a given right-angled plane triangle, let any line be drawn parallel to the base, and divided so that the square of the part adjacent to the hypothenuse may be always equal to the part adjacent to the perpendicular. It is required to determine the equation and area of the cur which is the locus of division. 130.-C is a given point, and RRR is a line given by position: now, if perpendiculars RP, RP, &c. be drawn, so that CR: CP = a given ratio, what is be locus of the point P? 131.-Given x + y : x2 - y2 :: 1:37 and xy= 10 }; required x and y. 132-A person bought a horse, which he afterwards sold for 24 guineas. and, by so doing, lost as much per cent, as the horse cost him. What sum did he cost? 133. What will a marble frustum of a cone come to, at 12s. per solid foot; the diameter of the greater end being 4 feet, that of the less end 1 feet, and the length of the slant side 8 feet? 134.-The diameter of a legal Winchester bushel is 184 inches, and its depth 8 inches; what must the diameter of that bushel be whose depth is 7 inches. 135.—Three men bought a tapering piece of timber, which was the frustum of a square pyramid, one side of the greater end was 3 feet, one side of the less 1 foot, and the length 18 feet; what must be the length of each man's piece, supposing they paid equally and are to have equal shares? 136.-Find the roots, and what relation must have place among the coefficients, of the equation x2. ax+b= 0, so that one root may be the nth power of the other. 137.-Find an arc of a circle, such that, if the continual product of the tangent, cotangent, secant, and cosecant, be divided by the continual product of the sine, cosine, versed sine, and coversed sine, the quotient shall be a minimum. 138.-To find the area and the sides of a rectangle of which the perimeter and the diagonal are given. 139.-Given the hypothenuse of a right angled triangle = a, and the ratio of the other two sides as mn; required its area. 140.-The base of a right-angled triangle being given≈ 24, and the difference between the perpendicular and hypothenuse = 12; required the other two sides. 141.-The area of a certain isosceles triangular field is eight acres and a half; query, the sides and angles, when the area of the inscribed circle is equal half the field. At a certain school, where there are 171 boys to be taught Latin, writing, and accompts, separately, at four shillings, six shillings, and nine shillings per quarter; and the master is desirous to have as much leisure time as possible; he desires to know how many boys he must teach of each class, so as to do the least work for the most money, supposing his labour proportionable to his prices. 143.-If a round cistern be 26.3 inches in diameter, and 52.5 inches deep; how many inches diameter must a cistern be to hold twice the quantity, the depth being the same? 144.-A may-pole, whose top was broken off by the wind, struck the ground at 15 feet distance from the bottom of the pole; what was the height of the whole may-pole, supposing the length of the broken piece to be 39 feet? 145.-What will the diameter of a globe be, when its solidity and superficial content are equal to each other, or rather when they are both expressed by the same number? 146.-In a right-angled triangle given the hypothenuse 30, and the difference between the base and perpendicular = 6; required those two sides. 147.-The area of a right-angled triangle being given equal to aa, it is required to determine the triangle, supposing the sides to be in geometrical proportion. va+va - x 149.-Given, in a plane triangle, the difference of the segments of the 18, and the sum of the sides 150; query, the dimensions when base 3F2 |
