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25-If the sum of the distances of the hours of three and four in the afternoon, from the meridian, or 12 o'clock line, be 82° 48'; query, the latitude for which this horizontal dial was made.
26.— The difference of longitudes of two places under the arctic circie being given = 52°; to determine the sun's declination when he begins to appear to the inhabitants of one place, at the same instant of time that he sets to those of the other.
27.-The fences of a piece of land are the abscissa, ordinate, and curve of the conic parabola. Now there is given the perimeter or sum of the three sides = 80 chains, and the ordinate is to the abscissa as 6 to 5: required the area by a simple equation.
28.-A gentleman bought a field in the form of a right-angled triangle, the sum of whose sides is 60 chains, at the following rate, viz. for every chain of the perpendicular he was to pay two guineas, and for each chain of the base one guinea. Query, the sides of the field where the gentle. man has the most land in proportion to his money.
29.-Suppose a globe of wood, when put in water, be observed to rise two inches above the surface of the water, and afterwards, when put into another liquor whose specific gravity to that of water is as .92989 to 1, it rises only 1.25 inch above the surface of the liquor: it is required to determine the globe's diameter.
30.—What angle must a projectile make with the plane of the horizoa, with a given velocity (a) per second, to describe in its flight the greatest area possible?
31.-A number of gentlemen at a tavern pay 6l. If there had been two more, each person would have paid 5s. Iess : required the number of gentlemen, and the portion which each gentleman paid of the reckoning.
32.- To divide a given angle into two parts, such that their sines may bave the given ratio of m to n.
33.- Required the value of x in the equation =f*-?.
34.-Given the side of the base of a square pyramid = 45 inches, and its altitude 21 times the diameter of a circle whose area is equal to that of the base of the pyramid: at what distance from the vertex must the same be cut (by a plane parallel to the base), that the solidity of the lesser pyramid may be to the solidity of the frustram in the ratio of 2 to 1?
35.-A gentleman, having a garden in the form of an ellipsis, whose transverse diameter is 300 and conjugate = 210 feet, is determined to raise the said garden-plot one foot higher, by a trench that he will make round it. Required the breadth and depth of the trench, supposing them equal.
36.-Given the perpendicular let fall from the right angle of a rightangled triangle = 55; to determine the area of the triangle, when the same is a minimum.
37.-An erect declining dial declines from the south 30 degrees, and the plane's difference of longitude exceeds the substyle's distance from the meridian just equal to the co-latitude of the place. To determine in what latitude this dial is fixed.
38.--It is required to find two numbers, such that the product of the greater and square root of the less may be equal to 48, and the product of the less and square root of the greater may be 36.
39.-In a mixture of rum and brandy, the difference between the quantities of each is, to the quantity of brandy, as 100 is to the number of gallons of rum; and the same difference is to the quantity of rum as 4 to ibe number of gallons of brandy. How many gallons are there of each?
40.-A market-woman, with a basket containing a certain number of eggs, sold half that number and half an egg to one person ; half the remainder and half ap egg to a second; half the remainder and half an egg to a third ; and then had one egg left. How many had she at first?
41.-To find the area of a plane triangle, when two of its sides and tho included angle are given.
42.-Reduce the surd V Tiitso to an equivalent; one having no denominator affected by the radical sign.
43.- A person baving doubled, at the gaming-table, the money which he bad before he began to play, gave a sovereign to his domestic. Gaining, a second time, enough to double the money which he bad remaining, he purchased, for a sovereign, a lottery-ticket, which came up a blank. Approaching the gambling-table for the third time, and doubling bis money, he found that he had only a sovereign remaining in his pocket. Required what moncy he bad at first?
44.—Required a vulgar fraction equivalent to the repeating decimal 0.142857142857, &c.
45.–Given to t y* = 130721 = a, x*ys = 3961758400000 = b; query, x and y.
46. —The diameters of an ellipsis are 60 and 40; what is the length and breadth of an oblong inscribed therein, whose content is 1152?
47.-In a right-angled plane triangular field, the legs are 3.r* and 20+;
The unknown depths of algebra explore,
Their product just a lady's age will show.
50.-A person paid a bill of 501, with.half-guineas and crowns, using in all 101 pieces. How many pieces were there of each sort? 51.-Required the value of 15o V6300 ?
v 52.- Required the value of ?
v b 53.- What quantity of canvas will be necessary for.forming a conical tent, whose height is 8 feet, and the diameter at bottom 13 feet?
54.-How many square feet of board are required to make a rectan. gular box, whose length is to be 31 feet, breadth 2 feet, and depth 20 inches ?
53.-Sappose that county contains one hundred thousand inhabia
tants, and that the population increases every year by a thirtieth part i what will be the number of inhabitants at the end of a century ?
56.-A cornfactor bought wheat at 5s. per bushel, and barley at 2s. per bushel : now, in laying out 100 guineas, be observed that, if the square of the number of the bushels of what were multiplied by the number o. the bushels of barley, the product would be a maximum. How many bushels of each sort did he buy?
57.-A gentleman, by will, left legacies to his four servants, A, B, C, and D, proportional to the time each had been in his service; but, ia case any of them should die before the expiration of the year, their share or shares to be divided equally among the otbers: accordingly, B died, and his share, so divided, made the share of C a mean proportional to those of A and D; whereas, before, A was to have 781., C 301. and D 6.. What money did each receive?
58.-Given the difference of the two parallel sides of a trapezoid = 5.642224 poles, and the area one acre: query each side when the perimeter is a minimum.
59.--It is required how many times a conical glass, whose depth is 4 inches, and its internal concave surface is double the area of its base, or brim, may be filled out of a punch-bowl, whose form is a parabolic conoid, its depth 12 inches, and its internal concave surface double the area of its base, (as in the cone,) by a quadratic equation.
60.-Required the contents of the solid generated by the revolution of the cissoidal area AMP about the axis AB? (See Figure, p. 428.)
61.-Express the surd V° 73° in its most simple form.
62.-The area of a right-angled triangle, whose sides are in arithme. tical progression, being given equal to 216; to determine the triangle.
63.-Required two numbers, such that twice their sum, three times their product, and the difference of their square, shall be equal to one another.
64.-Given the solidity of the sector of a sphere = 1413.72, and the radius = 15; to determine the greatest inscribed cylinder.
65.-A water-mill is to be built where the current of water has a fall of 22 feel, or perpendicular descent: the architect desires to know the diameter of the water-wheel, so that the power or force it shall receive from the issuing water may be the greatest possible !
66.-Required the perimeter of the least common parabola that will circumscribe a circle whose diameter is 32 inches.
67. What numbers are those which, being both multiplied by 27, the first product is a square, and the second the root of that square; but, being both multiplied by 3, the first product is a cube, and the second the root of that cube?
68.-A farmer bas two cubical stacks of bay; the side of one is 3 yards longer than the side of the other, and the difference of their contents is 117 solid yards. Required the side of each?
70.-Required the fluxion of x V (a + **)?
71.-What is the side of that equilateral triangle, whose area cost as much paving, at 8d. per foot, as the pallisading the three sides did at a guinea a yard?
72.-What would a circular reservoir, whose diameter at top is 40 yards, at bottom 38% 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.-Wbat 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 fot long, and 9 inches in compass, weighs 22 lbs. ; what will a fathom of that cable weigh whose diameter is y inches?
76.-Required that number, of which p times the mth power is equal to & times the (m + 2)th power? 77.–Given VIP}, required x and y.
V*+ y =7\' 78.-The convex superfices of the common parabolic conoid will bo least when its abscissa is 0.903084 X V(s); and its greatest ordinate 0.839605 x V (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 in,scribed square = 20.2825. Required the area and sides of the triangle.
80.--There is an isosceles triangle, whose side, being added to tho 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 ar gles by a simple equation.
82.-To find that fraction which excecds 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 (ax' + b)' +27(a' - *') x (x - ).
85.—Required the formula for summing an infinite scries 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 th“ garden, that, if walks be made from cach 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, BP and BC= 80 miles, making a right angle at B; and supposing a messenger sets out from X, 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 bour, until he arrives at D Now, supposing the angie EAD to be 35°, in what time will be arrive at C?
89.–Given the rectangle of two sides of a plane triangle = 1680, 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 cburch to be 6 feet in diameter, wbat would the gilding of it come to, at 3ed. per square inel?
92.--A person wants a cylindrical vessel of 3 feet, that shall hod teme as much as another of 28 inches deep, and 46 inches in diamoter: what must be the diameter of the required vessel?
93.-Two porters agreed to drink off a quart oi strong beer betrien thein, 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 touchd 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 wbich is 5.7 inches, the diameter at the top 3.7 inches, and that of the bottom 4.23 inches? 94.-Given u = log.
SV (1 +*2)+7(l— **)?
,sequired the value of u. 95.— Required the fusion of I \ V(1 + **)+*
IV (1+*')-* 96.—Required the fusion 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 (if inscribed and circumscribed circles ?
98.-In any right-angled triangle, the area (= 294), and the diference between the hypothenuse and perpendicular (= 14), being given; to ond 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 bappen'd, one day, at a tavern to be,
Carousing and drinking of port mighty free,