2. How many cubes of 24 inches in the edge can be cut from a cube of which the edge is 27 inches ? 3. How many cubes whose edges are 4 inch long can be contained in a box of which the base is 18 square inches, and the depth 71 inches? 4. What length must that solid be to contain a cubic foot which is 8 inches broad and 3 inches wide? 5. If a solid parallelopiped contain 100 cubic yards, and its base contain 100 square feet, find its height. 6. How many cubic yards of earth were dug from an excavation a mile and a half long, 30 yards wide, and 24 feet deep? 7. What is the number of cubic feet and inches in a piece of masonry 9 feet 3 inches long, 11 feet 5 inches high, and 3 feet 2 inches thick? 8. Find the length of a solid whose thickness is 1 foot, breadth 18 inches, and solid contents 3 cubic feet 216 inches. 9. What must be the length of a trench which is 5 feet deep and 91 feet wide, in order that the entire excavation may contain 7,040 cubic feet? 10. A rectangular cistern whose length is 13 feet, and breadth 6 feet, contains 2944 cubic feet. What is the depth? 11. How many bricks will be required to build a wall 20 yards long, 71⁄2 feet high, and 13 inches thick, supposing a brick to be 9 inches long, 4 wide, and 3 thick? If there be in the wall a doorway 6 feet 4 inches high and 4 feet wide, how many bricks less would be required? 12. A chest is 6 feet 2 inches long, 3 feet 4 inches wide, and 2 feet 9 inches deep, and is composed of boards 1 inch thick. Find the quantity of wood used, the contents of the chest, and the difference between its external and internal surfaces. XXVII. 1. If a cubic foot of wood weigh 12lb., what is the weight of a beam whose length, width, and depth are 24, 24, and 2 feet? 2. The weight of a cubic foot of Portland stone is 156 pounds; find the weight of the first stone of the Fitzwilliam Museum, at Cambridge, the mean dimensions of it being: length 7 feet, breadth 3 feet 9 inches, and thickness 2 feet. 3. What is the weight of a block of stone 12 feet 6 inches long, 6 feet 6 inches broad, and 8 feet 3 inches thick, when a block of the same stone 5 feet long, 3 feet 9 inches broad, and 2 feet 6 inches thick, weighs 7,500lb. ? 4. Find the weight of a rectangular vessel of iron an inch and a half thick without a top, the vessel being 10 feet 8 inches long, 8 feet 4 inches broad, and 5 feet deep, supposing a pound weight of iron to have a bulk of 4 cubic inches. 5. How many cubic feet of lead one-eighth of an inch thick would be required to line a cistern whose length, breadth, and depth are 7 feet 10 inches, 6 feet 3 inches, 3 feet 4 inches? 6. A cubic foot of copper weighs 4cwt. 3qrs. be drawn into a wire 1 mile 3 furlongs 5 poles long. copper requisite for a single wire of 50 miles. a section of the wire. 24lb. 4oz., and can Find the weight of Find also the area of 7. How long is an iron bar that contains a cubic foot of iron when its breadth is of an inch, and thickness is an inch? 3 4 19. 1 8. Find how many square yards a cubic foot of gold would cover when beaten out Too of an inch thick, and show that its extent is greater than 442, but less than 4 of a square mile. 1 1 441 9. If gold can be beaten out so thin that a grain will form a leaf of 56 square inches, how many of these leaves will make an inch thick, the weight of a cubic foot of gold being 10cwt. 95lb. Avoirdupois? XXVIII. 1. If 6 men earn £7 6s. 3d. in 7 days, how much will 10 men earn in 112 days? 2. If 4 men earn £4 7s. 6d. by working 5 days of 10 hours each, how much would 7 men earn in 3 weeks working 8 hours a day? 3. If 7 men and 10 boys earn £6 a week, and 3 men and 5 boys earn £11 in 4 weeks, in what time will 5 men and 4 boys earn £14? 4. The wages of A and B together for 7 days amount to the same sum as the wages of A alone for 129 days. For how many days will the sum pay the wages of B alone? 5. If 3 men do as much work as 5 women, and 2 women as much as 5 children; divide between 9 men, 11 women, and 6 children a day's wages amounting altogether to £3 5s. 1d. 6. The wages of one man, one woman, and three children amount to 30s. per week. The man and one child earn together 4 times as much as the woman. The man and three children earn 5 times as much as the woman. Find the wages of each. 7. If 20 men, 40 women, and 50 children receive among them £350 for 7 weeks' work, and 2 men receive as much as 3 women or 5 children, what sum does a man, a woman, and a child receive weekly? 8. Three men are employed on a work, working respectively 8, 9, 10 hours a day, and receiving the same daily wages. After 3 days each works one hour a day more, and the work is finished in 3 days more. If the total sum paid for wages be £2 7s. 64d., how much of it should each receive? XXIX. 1. If 100 men can perform a piece of work in 30 days, how many men can perform another piece of work thrice as large in one-fourth part of the time? 2. If 5 men or 7 women can do a piece of work in 37 days, how long will a piece of work twice as great occupy 7 men and 5 women? 3. If 12 men and 9 boys do a piece of work in 25 days, find how many men, assisted by an equal number of boys, would be required to do the same in 9 days, supposing 2 men to do as much as 3 boys. 4. If 3 men, 4 women, 5 boys, or 6 girls can perform a piece of work in 60 days, how long will it take 1 man, 2 women, 3 boys, and 4 girls, working together? 5. Men, women, and children are employed to do a piece of work. A man can do twice as much as a woman, and 6 times that a child; 8 women and 18 children can do the work in 40 days. How long would it take 10 men to do it? 6. Three men and two women take three days to do a piece of work which four women and three boys can do in 5 days, and one man does as much as three women in the same time. How long would it take a man, a woman, or a boy to do the work alone? 7. If 5 men and 2 boys can complete a piece of work in 6 days which 5 boys and 2 men will complete in 12 days, how long would a single man or boy take to complete it by himself? 8. If a man can do treble, and a woman double the work of a boy in the same time, how long would 9 men, 15 women, and 18 boys take to do double the work which 7 men, 12 women, and 9 boys complete in 250 days? 9. If 6 men and 2 boys can reap 13 acres in 2 days, and 7 men and 5 boys can reap 33 acres in 4 days, how long will it take 2 men and 2 boys to reap 10 acres? 1. Two persons, A and B, finish a work in 20 days, which B by himself could do in 50 days. In what time could A finish it by himself? How much more of the work is done by A than by B? 2. A alone can do a piece of work in 11 days, and B alone can do it in 17 days. How long would they take to do it together? 3. A and B do a piece of work together in 6 days, and B can do a fifth of the same in a day and a half. How long would each be in doing it alone? 4. A and B can build a wall together in 12 days, A and C in 15, Band C'in 20. In how many days can they do it (1) working all together; (2) each working separately? 5. A and B can do a piece of work alone in 15 and 18 days respectively; they work together at it for 3 days, when B leaves, but ▲ continues, and after 3 days is joined by C, and they finish it together in 4 days. In what time would C do the work by himself? 6. A and B can perform a work in 7 days, A and C in 8 days, and A, B, and C in 5 days. In what time can B and C perform it? 7. A does of a piece of work in 4 hours, B does 2 of what remains in 1 hour, and C finishes it in 20 minutes. How long would they have been in doing the whole if they had worked together? 8. If A can perform a work in 9 days, B in 10, and C in 11, in what times could A and B, A and C, and B and C respectively complete the work? 9. A, B, and C can each complete a certain work in 9, 12, and 18 days respectively. If A and B work together for 2 days, and A and C afterwards for 2 more, how long will B and C working together take to finish the work? 10. If A can do as much work in 5 hours as B can do in 6 hours, or as C can do in 9 hours, how long will it take A to complete a piece of work, one half of which has been done by B working 12 hours, and C working 24 hours? 11. If A in 2 days can do as much work as C'in 3 days, and B in 5 days as much as C in 4 days, how long will B require to execute a piece of work which A can accomplish in 6 weeks? 12. A is twice, and B is just one and a half times as good a workman as C. The three work together for two days, and then A works on alone for half a day, and B for one day. How long would it have taken A and C together to complete as much as the three will have thus performed? 13. If A can by himself perform a certain quantity of work in 5 days, B twice as much in 7, and C 4 times as much in 11 days, in what time can A, B, and C together perform 3 times the original quantity? 14. A can do as much work in 2 days as B in 3 days, and B as much in 4 days as C in 5 days. In what time could A, B, and C together do a piece of work which A can do in 11 days? XXXI. 1. If a cubic foot of water weigh 1,000 ounces avoirdupois, find the number of grains Troy in a cubic inch of water. 2. A rectangular tank is 13 feet 6 inches long by 9 feet 9 inches wide. How many cubic yards of water must be drawn off to make the surface sink 1 foot? 3. How many tons of water will a cistern contain whose length, breadth, and depth are respectively 18 feet 8 inches, 10 feet 4 inches, and 6 feet 9 inches? What is the amount of pressure of the water on the bottom of the cistern? 4. If 13 cubic feet make 81 gallons, how many hogsheads will be contained in a cistern 42 feet long, 39 feet broad, and 10 feet deep? 2 5. A cistern is full of water, and after 35 gallons are drawn off it is found to be full. How many gallons does it contain? 6. Two equal wine-glasses are filled with a mixture of spirit and water, one containing 3 parts of water and one part of spirit, and the other 4 parts of water and 3 parts of spirit. When the contents of the two glasses are mixed in a tumbler, find how many parts of the whole mixture are wine and water. 7. Four gallons of wine and one of water are poured into a vessel, and one gallon of the mixture is poured out into a second vessel; then one gallon of water is poured into the first vessel, and the process is performed four times. Determine the quantity of wine in each vessel. 8. If a cubic foot of atmospheric air weigh an ounce and a quarter under the ordinary pressure, what would be the weight of air contained in a room 30 feet long, 203 wide, and 10 high? 9. If the pressure of the atmosphere on a square inch at the surface of the earth be about 14 pounds 11 ounces avoirdupois, what is the pressure on a square foot and on a square yard? 10. What is the pressure on a square foot at the bottom of a pond 20 feet deep, if the weight of a cubic foot of water be 1,000 ounces? If the pressure of the atmosphere be taken into account, what is the total pressure? 11. If the weight of a cubic foot of water be 1,000 ounces avoirdupois, and mercury be 131⁄2 times as heavy as water, find the height of a column of water which shall be equal to the pressure of mercury in the barometer when it stands at 30 inches. XXXII. 1. If a vessel of water be emptied in 5 hours through a spout A, and in 3 hours through a spout B, in what time will it be emptied through both spouts together? 2. If a cistern when full of water can be emptied in 15 minutes by a pipe, and when empty can be filled by another in 20 minutes; if the cistern be full, in what time can it be emptied by both pipes being opened at the same time? 3. If a pipe which conveys 13 gallons in a minute, lower the surface of water in a tank 11⁄2 inches in 45 minutes, in what time will a pipe which conveys 4 gallons in a minute lower the surface 5 inches ? 4. A cistern is filled by two spouts in 20 minutes and in 24 minutes respectively, and emptied by a tap in 30 minutes; if the cistern be empty, what portion of the cistern will be filled in 15 minutes, when all three are opened together? 5. A cistern has 3 pipes, A, B, C; A will fill it in 3 hours, B will fill it in 4, and C will empty it in 1 hour. The cistern being empty, these pipes are opened at 1, 2, and 3 o'clock respectively. At what time will the cistern be full or empty, and which? 6. In what part of a day will four fountains, being opened together, fill a cistern, which if severally opened, they would each fill it in one day, half a day, the third and the sixth part respectively ?— (Lilavati). |