461. The area of the surface of a sphere is equal to four times the area of a great circle of a sphere. 462. The volume of a sphere is equal to one-third of the product of the area of the surface by the radius. FORMULAS. 463. If V stands for volume, S for lateral surface, B for base, T for total surface, P for perimeter of base, R for radius of base, H for altitude, and L for slant height, the formulas for surfaces and volumes will be as follows: 1. Find the total surface of a rectangular parallelopiped whose length is 15 ft., breadth 9 ft., height 6 ft. 2. Find the total surface of a triangular prism whose height is 6 ft., side of base 3 ft. 3. Find the volume of a prism whose base is a square 8 ft. on a side, and whose length is 40 ft. 4. Find the volume of a right prism 32 ft. long, if its ends are trapezoids the parallel sides of which are 12 ft. and 8 ft., and the perpendicular distance between these parallel sides is 6 ft. 5. A right cylinder is 10 ft. high, and measures 7 ft., 4 in. around the base. Find the lateral surface and the volume. (=34.) 6. Reckoning 7 gallons to the cubic foot, if 375 gals. are pumped out of a round cistern 7 ft. in diameter, how many inches will the surface of the water fall in consequence? 7. A marble column measures 7 ft. 4 in. in circumference and is 15 ft. long. Find the expense of polishing the entire surface at $1.50 a square foot. 8. Find the total surface of a regular pyramid when each side of its triangular base is 6 ft. and its slant height is 18 ft. 9. Find the volume of a regular pyramid whose base is an equilateral triangle measuring 4 ft. on a side, and whose height is 15 ft. 10. How many square feet of canvas is required for a Iconical tent the altitude of which is 8 ft. and the' diameter of the base is 7 ft.? (= 3.1416.) 11. A conical tent whose slant height is 12 ft. requires 132 sq. ft. of canvas to cover it. Find how many feet of ground the tent covers. (π = 34.) 12. Find the volume of a right cone the height of which is 15 ft. and the circumference of the base 14 ft. F 13. Find the total surface of the frustum of a pyramid 14. How many square feet of tin will be required to make to be 28 in. and 14 in. respectively, and the height 15. Find the volume of the frustum of a regular square pyramid whose height is 24 ft., and the sides of its 216. Find the volume of the frustum of a right cone if the 17. Find the surface of a sphere whose diameter is 10 in. 19. The ball on the top of St. Paul's cathedral in London inch? 20. Find the volume of a sphere if the diameter is 14 in. 21. Find the volume of a sphere whose circumference is 22. How many gallons of water will a hemispherical bowl F CHAPTER XVI. MISCELLANEOUS PROBLEMS. DECIMAL FRACTIONS. 1. Four men together paid $20,000 for some land. The first puts in $2350, the second $5820.35, the third $7640.75. How much must the fourth man pay? 2. What will be the cost of uniforms for a base-ball nine at $2.87 for each uniform? 3. At $15.87 a ton, what will be the value of 637 tons of hay? 4. If peaches are worth $1.25 a basket, and it takes 27 dozen for a basket, what is the value of 26,028 dozen peaches? 5. If 964 baskets of peaches are sold for $1301.40, what is the price per basket? 6. If 324 men contribute together $2647.08, what is the contribution of each? 7. A boy picks blueberries in a pasture, giving to the owner of the pasture for the privilege 1 quart out of every 8 quarts. In 2 days he picks 48 quarts, and sells his share of the berries for $3.78. What did he get a quart? 8. If 150 men work on a railroad at the same price per day, and if, at the end of the week, they all together receive $1575, what price per day does each man receive? 9. A boy has 3 pieces of twine; one is 58.74 ft. long, another is 97.86 ft., and a third 57.26 ft. How long a kite-string can he make? 10. If a kite-string is 213.86 ft. long, and the kite breaks away and carries off 94.38 ft. of the string, how much will be left? How much more must be bought to make up 1000 ft.? 11. One man has in his purse 3 hundred-dollar bills, 16 ten-dollar bills, 12 one-dollar bills, and 19 ten-cent pieces. Another man has in his purse 13 ten-dollar bills, 63 five-dollar bills, 8 one-dollar bills, 2 halfdollars, 3 quarters, 4 five-cent pieces, and 17 cents. How much more money has one man than the other? 12. If 12 boys play hare and hound, and run 2.876 miles, find the whole number of miles run by all the boys. 13. A boy sold 19 dozen eggs at 26 cts. a dozen, 25 lbs. of wool at 35 cts. a pound, and 13 chickens at half a dollar apiece. He bought two sheep at $5.75 each. How much money had he left? 14. If 1964.52 bu. of corn are to be put into bags holding 2.14 bu. each, how many bags will it take? 15. At $9.17 a barrel, how many barrels of flour can be bought for $876.35, and how much money will be left over? 16. Boys in playing hare and hound run 3.876 miles. The hares drop a piece of paper every 4.75 feet on the average. How many pieces do they drop? 17. If a man earns $3.25 a day, how many days will it take him to earn $1964.87? |