2. Find the formula for the area of a the diameter instead of the radius. circle in terms of 3. (a) Most modern houses and buildings are heated by passing steam or hot water through a set of cylindrical pipes called a radiator. The surface of these pipes is called the radiating surface, and its size depends upon the size of the room. It is measured in square feet. (b) Find the amount of radiating surface from 12 pipes, 2" in diameter and 35" high. (Disregard the bases.) (c) A radiator has two rows of ten 2" pipes 32" high. 4. (a) The amount of water Ο of the circle inside the pipe. (b) Find the area of a cross section of a pipe whose inner diameter is 8". (c) The thickness of the iron of this pipe is 1". What is the diameter of the outer cross section? Find its area. (d) From these two areas how may the area of the ring be found? 5. There is an easier way to find the area of a ring. Let the outer circle be 1 with a radius of r1 or 5, and the inner circle be ✪ 2 with a radius of r1⁄2 or 4. 6. (a) The radius of the inner cross section of the first pipe is 3 in. and of the second 2 in. How .. 1 is 24 times as large as 2. (c) Do you have to compute the exact areas of two circles to find the ratio of their areas? (d) The areas of two circles have the same ratio as the squares of their radii or as the squares of their diameters. Show why radii or diameters may be used in these ratios. 7. (a) The rate of the flow of water through a cylindrical pipe is proportional to the area of its cross section. (b) Two pipes have 1" and 2" inside measurements, respectively. Then O 1 2 = = 12 22 1 4 Therefore 4 times as much water will flow per minute through the second pipe as through the first. 8. How much more water will flow per minute through a 3" pipe than through a 1" pipe? 9. How much faster will water flow through a 4" pipe than through a 11⁄2" pipe? 10. In a park is a fountain in the center of a circular grass plot 300 ft. in diameter. A 10-ft. cement walk surrounds the plot. What is the area of the walk? 11. (a) If the equatorial diameter of the earth is 7924 miles, how big is the equator? (b) How many miles long is 1° at the equator? 12. (a) A company advertises for bids for painting 9-ft. bands around telephone poles whose average diameter is 14′′. (b) Mr. A figures paint at $3.00 per gallon and allows 1 gallon to 275 sq. ft. He figures 1 hour's time for painting a square and 7 hours' extra time per C poles for moving material from one to another. The labor costs $.70 per hour. (c) Mr. B makes a bid of $65 per C poles, with all materials furnished. (d) To which man should the company give the contract? How much is saved thereby? 13. (a) An oatmeal box is 7" high and has a diameter of 41". How large must be the paper used for the label around it? (b) How much cardboard is needed for the box? 14. (a) A box of Dutch Cleanser is 43" high. Its diameter is 3". How large is the label covering the side? (b) How large is the tin in each end? VOLUME A. VOLUME OF A PRISM 1. A cube 1 cm. long is a cubic centimeter. A cube 1' long is a cubic foot. 3. (a) How many 1" cubes can be laid on the bottom of a cubical box 1 ft. long? (b) It is evident that there can be one cube for each square inch of surface of the bottom. (c) Since 1'= 12", there can be 12 x 12 or 144 cubes in one layer. (d) Since the cube is 12" high, how many layers of cubes can be put in? (e) Evidently the total number of small cubes in the box is 12 times 144 or 1728. Therefore 1 cu. ft. 123 = 1728 cu. in. = 4. Suppose the box were only 8" high. The number of cubes would be 8 x 12 x 12. 5. Suppose the box were 9" wide. Then each layer would have 12 x 9 cubes. 6. If the box were 12" long, 9" wide, number of cubes would be 12 × 9 × 8. rectangular prism. and 8" high, the Such a box is a 7. The number of cubic inches the box can contain is called its volume. |