Find the areas of circles with the following radii and circumferences : 7. 5 in., 31.416 in. 8. 4 in., 28.2744 in. 9. 4 in., 25.1328 in. 10. 31⁄2 in., 21.9912 in. 11. 5 in., 34.5576 in. 12. 10 in., 62.832 in. Find the areas of circles with the following radii : 17. 3 in. 20. 20 in. 23. 0.1 in. 26. in. 29. 2.3 in. 30. What is the circumference of a water pipe that is 4 in. in diameter ? 31. What is the area of a cross section of a water pipe that is 3 in. in diameter ? 32. A water pipe is 4 in. in external diameter. If the iron is in. thick, what is the internal circumference? 33. A school globe is 15 in. in diameter. How long is the equator on the globe? 34. The equator on a school globe is 62.832 in. What is the radius of the globe? 35. What is the circumference of the largest circle that can be drawn in a square 10 ft. on a side? 36. What is the area inclosed between the circle and the sides of the square in Ex. 35? 37. What is the diameter of a circular plot of ground of which the circumference is 691.152 ft.? What is the area? 38. A pipe is 3.75 in. in external diameter. If the iron is in. thick, what is the area of an internal cross section? 39. A pipe has an internal diameter of 3.5 in. The iron is in. thick. What is the area of an external cross section? 121. Cylinder. A solid formed by the revolution of a rectangle about one of its sides is called a cylinder. The two circles that are thus formed are called the bases, and the curved surface is called the lateral surface of the cylinder. The distance between the two bases is called the altitude of the cylinder. 122. Volume of a Cylinder. We can think of 1 cu. in. on each square inch of the base. Therefore, if the base contains 4 sq. in., and if the cylinder is 1 in. high, the volume will be 4 x 1 cu. in., or 4 cu. in., and if the cylinder is 5 in. high, the volume will be 5 × 4 cu. in., or 20 cu. in. Therefore, to find the volume of a cylinder, Multiply the number of cubic units corresponding to the number of square units of the base, by the number of units of altitude. That is, if the base has 9 sq. in., and the altitude is 12 in., the volume is 12 x 9 cu. in., or 108 cu. in. If the base has a diameter of 2 in., and the altitude is 5 in., then the area of the base is 3.1416 × 1 sq. in. (§ 120), and the volume is 5 x 3.1416 x 1 cu. in., or 15.708 cu. in. 123. Lateral Area of a Cylinder. We may think of the surface of a cylinder as unrolled, like a paper reaching once around a pencil. In this case it would unfold into a rectangle, one side of which would be the circumference of the base and the other the altitude. Therefore, to find the lateral area of a cylinder, Multiply the number of units in the circumference by the number of units in the altitude. For example, if the circumference is 6 in. and the altitude 4 in., the number of square units of lateral area is 4 × 6, or 24. We may express the work thus: Find the areas of base of cylinders whose volumes and altitudes are as follows: 9. 324 cu. ft., 18 ft. 10. 323 cu. ft., 17 ft. 11. 28 cu. ft. 288 cu. in., 2 ft. 2 in. 12. 81 cu. ft. 576 cu. in., 5 ft. 4 in. Find the volumes of cylinders whose altitudes and radii of base are as follows: 13. 27 in., 4 in. 14. 25 in., 6 in. 15. 32 in., 8 in. 17. 2 ft. 6 in., 2 in. 18. 3 ft. 4 in., 1 ft. 124. Capacity of Cisterns. The following measurements are used in computing the capacity of cisterns: 1 gal. = 231 cu. in. 1 cu. ft. 7.48 + gal. = nearly 7 gal. 31 gal. = 1 bbl. = nearly 4 cu. ft. In Exercise 65 use 71⁄2 gal. for 1 cu. ft., 43 cu. ft. for 1 bbi., and 34 as the multiplier in cases requiring the area of base, given the diameter. Find the number of gallons of water in a cylindrical tank 40 ft. in diameter, the water being 35 ft. deep. Find the number of gallons in these rectangular tanks: Find the number of barrels in the following tanks: 7. 12'x11' x 9'. 8. 15'x14' x 8'. 9. 15×12' × 52'. Find the number of gallons in cylindrical tanks of the following diameter and depth of water respectively : Find the number of gallons of water in wells of the following diameter and depth of water respectively : 125. Brick Work and Stone Work. The number of bricks required for walls is estimated by the thousand, and 22 bricks of ordinary size laid in mortar are allowed to each cubic foot of wall. A brick of ordinary size is 2′′ × 4′′ x 8′′, and its volume is therefore 64 cu. in., or cu. ft. Allowing for mortar, however, it is reckoned cu. ft., as stated. as Stone work is reckoned by the cubic foot or cubic yard, and sometimes by the perch. A perch is 161 × 1 × 1', and therefore equals 24 cu. ft. Practically, however, it is taken as 25 cu. ft., although this varies. In estimating the labor for brick and stone work, masons measure the outside of the walls, thus reckoning the corners twice. This is considered fair because of the extra trouble in laying a corner. Openings are generally not considered, although sometimes an allowance for half of the openings is made, the gain to the mason being offset by the extra labor in building around them. In estimating the material they will need, however, masons consider the corners and openings. The custom varies in these matters. For example, find the number of bricks required for a wall 42' x 25' and 1 ft. thick. How many cubic feet should be allowed in estimating the labor on a cellar wall 40' x 30', 6' high, and 2' thick? Perimeter of cellar, outside the wall, is 2 × (40 + 30) ft., or 140 ft. Therefore the number is 6 x 2 x 140 cu. ft., or 1680 cu. ft. EXERCISE 66 Find the number of cubic feet in walls of these dimensions: 1. 62 ft., 4 ft., 1 ft. 2. 78 ft., 5 ft., 1 ft. 6 in. 3. 49 ft., 5 ft., 1 ft. 8 in. 4. 75 ft., 6 ft., 2 ft. |