296. Surface of a Sphere. Consider a sphere placed in a cylindrical box which is just large enough to receive it. The diameter of the cylindrical box and of the sphere will be the same, and the height of the box will equal the diameter of the sphere. It is a curious fact that the total surface of this sphere will be exactly equal to the lateral area of the box. The lateral area of the box equals circumference X altitude. If r = radius of box then the circumference equals 2 π × r. Since the altitude twice the radius we have, Lateral area = 2 π XrX 2 r = 42. This formula also gives the surface of a sphere. = 297. Volume of a Sphere. A sphere may be regarded as made up of a large number of small pyramids, all having their bases in the surface of the sphere and their vertices at its center. The sum of the bases of these pyramids is the surface of the sphere and their altitude is the radius of the sphere. Denoting the surface of the sphere by S and its volume by V, we have, S = 4 πr2, V = πr3. PROBLEMS 1. Find the surface of a sphere whose radius is 10 inches. 2. At $1.40 a square foot, find the cost of gilding the hemispherical dome of a cathedral if the diameter of the dome is 80 feet. Suggestion. Find half the surface of a sphere whose radius is 40 feet. 3. The diameter of the earth is approximately 8000 miles. Assuming it to be spherical in shape, find its total area. 4. Find the volume of a sphere whose radius is 5 inches. 5. If cast iron weighs 450 lb. a cubic foot, find the weight of a solid spherical cast iron ball whose radius is 6 inches. 6. Find the volume of the earth in cubic miles. (See problem 3.) UNITED STATES LAND SURVEYS 298. In 1802 Colonel Mansfield, the surveyor of the Northwest Territory, inaugurated the plan for surveying the public lands which is still in use. The general features of the plan were as follows: The entire public domain was first divided into parts called land districts. In each district a meridian line, called the principal meridian, was run through the entire district from North to South, and from some point on this meridian an East and West line was run which was called a base line. Parallel to the principal meridian, and to the base line, lines were run six miles apart, dividing the land into squares six miles on the side. These squares are called townships. A row of townships extending north and south is called a range. The ranges are designated as east or west. Thus, in Figure 1 the range of townships marked a, b, c, d, e, is range 3, East. The range containing townships a', b', c', d', e', f', g', is range 5, West. The township marked c is township 3 North, range 3 East. The township marked e' is township 2 South, range 5 West. Each township is divided into 36 square mile plots, called sections. They are numbered as shown in Figure 2, the section numbered 1 being the north-east corner of the township. Section 14 in the township marked b' would be described as section 14, Township 2 North, range 5 West. Added to this, there would be a designation giving the land district in which the section is located. The land surveyed in this manner consists of nearly all the land now in the United States which did not belong to the thirteen States at the time of the Revolution, together with the lands which were later ceded to the United States by these original States. A section of land is divided into smaller parts, as shown in the figure. Thus the piece marked S.E. of S.E. is called the south-east quarter of the south-east quarter. PROBLEMS N.E. N.W.1 (North West (North East Quarter) Quarter) N. of S.W. S.E. of S.E. 1. Make a figure representing a township, and on it mark the following: the S.E. of section 18; the West NW N NE Χ y of section 27; the S.W. of the N.E. of section 9. 2. Describe the piece of land marked X in this figure. Also describe the piece marked y. 3. At $75 an acre, what is the value of the South of the N.E. of section 1 ? 4. At 40¢ a rod what is the cost of fencing in a section of land? A quarter section? 5. A piece of land 100 rods wide and 240 rods long is divided into four equal pieces by two fences running through it. At 50¢ a rod how much will it cost to fence the land, including the partition fences and the fences around it? 6. What fractional part of a section is the N.W. of the N.W. ? If the whole section is worth $20,000, what is this part worth? How much will it cost to fence it off from the rest of the section at 45¢ a rod? 7. If I own the S.W. of section 4, how many acres do I own? Make a diagram representing the section and showing the piece just described. Divide the rest of the section into one half section and eighths of a section. 8. Find the number of acres in each of the pieces shown on the figures on this page. SIMILAR FIGURES 299. Definition of Similar Figures. Figures having the same form or shape are said to be similar. 300. Sides of Similar Figures Form Proportion. Above are shown three pairs of similar figures. One of the essential properties of two similar figures is that corresponding sides form a proportion. These are special cases of a very important property of all similar surfaces. 301. Areas of Similar Figures. The ratio of the areas of similar surfaces equals the square of the ratio of their corresponding lines. Thus, the ratio of the areas of two circles equals the ratio of the squares of their radii, or of their diameters. |