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5. Base is 20 ft. and hypothenuse 25 ft. ? 6. Altitude is 32 ft. and hypothenuse 40 ft. ? 7. Altitude is 30 ft. and hypothenuse 60 ft. ? 8. Base is 90 ft. and hypothenuse 100 ft. ? 9. Base and altitude are each 1 ft. ? 10. If I travel east 12 hours at the rate of 4 miles per hour, and then travel north 12 hours at the rate of 54 miles per hour, how far shall I be from the starting-point ?
11. What is the diagonal of a square measuring 50 ft. on a side ?
12. What is the side of a square the diagonal of which is 50 ft. ?
13. A rectangle is as wide as it is long, and contains 768 sq. ft. What is the length of its diagonal ?
14. What is the side of the greatest square that can be inscribed in a circle 20 ft. in diameter ?
Note.—The diameter of the circle will equal the diagonal of the inscribed square.
15. What is the side of the greatest square that can be inscribed in a circle 50 ft. in diameter ?
16. What is the distance from the lower corner to the farthest upper corner of a hall 40 ft. long, 30 ft, wide, and 18 ft. high ?
17. A church-spire, 180 ft. in height, stands on a plain. North of the spire, and at a distance of 300 ft. from its summit, is a stake. Measuring east from this stake a distance of 400 ft., I find that I am just 80 feet from the foot of a column which is directly between me and the spire. Allowing that the column is 15 ft. high, how many feet is it from the top of the column to the top of the spire ?
(a.) A SPHERE is a solid bounded by a curved surface, every part of which is equally distant from a point within, called the centre. (Fig. 1, p. 240.)
(b.) A line drawn from the centre of a circle to the surface is called a Radius, and a line drawn from any point in the surface through the centre to the opposite point is called a DIAMETER.
(c.) A Prism is a solid having its lateral faces parallelograms and its bases equal and parallel polygons. (See Fig. 2.) Note. – A Cube (defined in 25, c) is a kind of prism.
(d.) A CYLINDER is such a solid as would be formed by revolving a rectangle about one of its sides. It has also been defined to be “a round body with circular ends.” (See Fig. 3.)
(e.) A Pyramid is a solid bounded laterally by triangles, of which the vertices meet at a common point, and the bases form the sides of a polygon, which is called the base of the pyramid. (See Fig. 4.)
(f.) A Cone is a solid which has a circular base, and tapers regularly to a point called the vertex. (See Fig. 5.)
(g.) A Frustum of a cone or of a pyramid is a part cut off by a plane parallel to its base. (See Fig. 6.)
(h.) SIMILAR SOLIDS have the same shape, i. e. the angles of one of them equal the corresponding angles of the other, and the sides about the equal angles are proportional.
(i.) All spheres are similar. Two cones or two cylinders are similar when their altitudes are to each other as the radii or diameters of their bases.
(j.) The SURFACE of a sphere equals the square of its diameter multiplied by 3.1416.*
ILLUSTRATION. — The surface of a sphere 4 inches in diameter = 3.1416 times 16 sq. ft. = 50.2656 sq. ft.
(k.) The SURFACES OF SPAERES are to each other as the squares of their radii or diameters.
ILLUSTRATION. - The surface of a sphere 2 inches in diameter is of the surface of a sphere 5 inches in diameter.
(1.) The SOLIDITY, or Solid CONTENTS, of a sphere equals the product of the surface multiplied by } of the radius, or by $ of the diameter, or it equals f of the cube of the diameter multiplied by 3.1416.*
ILLUSTRATION.— If the diameter of a sphere is 12 feet, its surface will equal 3.1416 times 144 sq. ft., and its solid contents will equal $ of 12 times as many cubic feet; or 2 X 3.1416 X 144 cu. ft. Or the solid contents will equal $ of 123 times 3.1416 = 3 of 1728 times 3.1416 = 288 times 3.1416 cu. ft. = 904.7808 cu. ft.
(m.) The solidities of spheres are to each other as the cubes of their radii or diameters.
ILLUSTRATION. — A sphere of 3 ft. radius contains 27 as many cubic feet as a sphere of 2 ft. radius.
(n.) The solidities of similar solids are to each other as the cubes of their like dimensions.
ILLUSTRATION.- A cube 2 ft. on a side is to a cube 5 ft. on a side as 22 is to 5*, or as 8 is to 125.
(0.) The solidity of a prism or of a cylinder equals the product obtained by multiplying the area of its base by its altitude.
Illustration. If the base of the prism in Fig. 2, or of the cylinder in Fig. 3, contains 8 sq. ft. and the altitude is 5 ft., the prism and the cylinder each contain 40 cu. ft.
(p.) The convex surface of a cylinder is equal to the product obtained by multiplying the circumference of its base by its altitude.
* Soe foot-note, page 235.
ILLUSTRATION.-If the circumference of the base of the cylinder in Fig. 3 equals 9 ft., and the altitude of the cylinder equals 4 ft., the convex surface must equal 4 times 9 sq. ft. = 36 sq. ft.
(q.) The solidity of a cone or of a pyramid equals the area of its base multiplied by $ of its altitude.
(r.) The solidity of a frustum of a cone, or of a pyramid, equals the product of of its altitude multiplied by the sum obtained by adding the areas of its upper and lower bases to the mean proportional between those areas.
Nore.—The mean proportional of two numbers is the square root of their product. Thus, the mean proportional of 4 and 9 = V4X 9 = 6.
ILLUSTRATION.-If the altitude of the frustum of a cone or of a pyramid is 9 ft., and its upper base contains 18 sq. ft. and its lower base 32 sq. ft., we should find its solid contents by finding the product of 18 + 32 + ✓ 18 X 32 multiplied by $ of 9 = (12 + 18 + 24) X 3 = 54 X 3 = 162 cu. ft. (s.) What is the solidity of 1. A sphere 8 ft. in diameter ? 2. A sphere of 5 ft. radius ? 3. A sphere 20 ft. in circumference ? 4. A sphere 10 ft. in circumference ? 5. A cone 6 ft. high, with a base 3 ft. in diameter ? 6. A cone 9 ft. high, with a base 5 ft. in diameter ? 7. A cylinder 8 ft. high, with a base 4 ft. in diameter ? 8. A cylinder 12 ft. high, with a base 1 ft. in diameter ? 9. A prism 10 ft. high, whose base contains 17 sq. ft. ? 10. A prism 15 ft. high, whose base contains 20 sq. ft. ? 11. A prism 18 ft. high, whose base contains 18 sq. ft. ?
12. The frustum of á cone 6 ft. high, 3 ft. in diameter at the upper base, and 4 ft. at the lower ?
13. The frustum of a pyramid 10 ft. high, containing 20 sq. ft. in the upper base, and 30 sq. ft. in the lower ?
14. What is the diameter of a sphere containing 100 cu. ft. ?
15. What is the surface of a sphere containing 50 cu. ft. ?
20. A sphere of wood 3 inches in diameter weighs 2 lb. What will a sphere of wood 9 inches in diameter weigh?
21. How many spheres 1 inch in diameter will be required to balance a sphere 10 inches in diameter ?
22. How many cubic inches of iron are there in a hollow sphere 1 ft. in diameter, if the iron is 1 inch thick ?
SUGGESTION. — There will be as many cubic inches of iron as there are in the difference between a sphere 12 inches in diameter and one 10 inches in diameter.
23. How many cubic inches of iron are there in a spherical shell 14 inches in diameter and 2 inches thick ?
144. Arithmetical Progression.
(a.) A SERIES OF NUMBERS IN ARITHMETICAL PROGRESSION, or an ARITHMETICAL SERIES, is a series of numbers, each of which differs from the preceding by the same number.
(b.) An arithmetical series is obtained by continually adding the same number to, or subtracting it from, any given number. ILLUSTRATIONS. — We should have —
By adding 3's to 1............................ 1, 4, 7, 10, 13, 16, etc.