Volume of Solids 614. 1. What is meant by the volume of a solid? 2. How can you find the volume of a rectangular prism (§ 250) from its three dimensions? 3. Find the volume of a rectangular prism whose base is 5 feet by 4 feet and whose altitude is 5 feet. 615. To find the volume of a prism or a cylinder. 1. If a triangular prism with a base of 9 square inches were 1 inch high, how many cubic inches would it contain? if it were 2 inches high? 3 inches? 4 inches? 2. If a cylinder with a base 9 square inches in area were 1 inch high, how many cubic inches would it contain? if it were 2 inches high? 4 inches high? 4 3 2 3. How, then, can you find the volume of a prism or a cylinder from its altitude and the area of its base? 616. The volume of a prism or a cylinder is equal to the product of its altitude and the area of its base. 617. 1. What is the volume of a prism with a base 5 inches square, if its altitude is 3 inches? 2. Find the volume of a cylinder whose diameter is 8 decimeters and altitude 2 meters. 3. What is the capacity in bushels of a bin 8 feet square and 9 feet high, inside measurements? 4. How many gallons of water will a cylindrical vat hold, if it is 7 feet in diameter and 8 feet high? 5. Find the value at 63 a bushel of the wheat that would fill a bin 15 feet square and 12 feet deep. 618. To find the volume of a pyramid or a cone. 1. Take a hollow pyramid and a hollow prism of the same base and altitude. Fill the pyramid with sand and empty it into the prism. How many times must you fill and empty the pyramid of sand to fill the prism? What is the relation, then, of the volume of the pyramid to that of the prism? 2. Try the same experiment with a cone and a cylinder of the same base and altitude. What is the relation of the volume of the cone to that of the cylinder? 3. It is proved also in geometry that the volume of a pyramid or a cone is, respectively, one third that of a prism or a cylinder having the same base and altitude. 4. Since the volume of a prism or a cylinder is equal to the product of its altitude and the area of its base, how can you find the volume of a pyramid or a cone? 619. The volume of a pyramid or a cone is equal to one third the product of its altitude and the area of its base. 620. 1. What are the solid contents of a cone, the diameter of whose base is 6 feet and whose altitude is 9 feet? 2. Find the solid contents of a pyramid whose base is 10 meters square and whose altitude is 20 meters. 3. If a cubic foot of granite weighs 165 pounds, what is the weight of a granite cone the diameter of whose base is 6 feet. and whose altitude is 8 feet? 4. A pile of coal in the shape of a cone is 30 feet high and 132 feet in circumference. Find its volume. 5. What is the weight of a marble pyramid whose base is 4 feet square and whose altitude is 8 feet, if a cubic foot of marble weighs 171 pounds? 6. The pyramid of Cheops has a square base 746 feet on a side and an altitude of 480 feet. Find its volume. 621. To find the volume of a sphere. 1. As indicated in the figure, a sphere may be divided into a great number of figures that are essentially pyramids. The sum of the bases of these pyramids is the convex surface of the sphere, and the altitude of each pyramid is the radius of the sphere. Then how may the volume of a sphere be found? 2. It is proved also in geometry that the volume of a sphere is equal to the product of its radius and its convex surface. 3. Since (§ 611) 4r2 represents the convex surface of a sphere, and r its radius (§ 619), §r × 4πr2, or § πr3, represents the volume of a sphere. 622. The volume of a sphere is equal to one third the product of its radius and its convex surface, or r3. 623. 1. Find the volume of a sphere whose radius is 7 feet. 2. What is the volume of a sphere 21 centimeters in diameter? 3. The circumference of a sphere is 22 inches. Find its volume. 4. A spherical aquarium has a diameter of 14 inches, measured on the inside. How many cubic inches of water will it hold? Find the weight of water that it will hold. 5. Find the weight of a 6-inch cannon ball made of iron that has a specific gravity of 7.21. Similar Surfaces 624. Figures that are of exactly the same shape though they differ in size are called similar figures. All circles are similar; also all squares, and all regular polygons having the same number of sides. Two maps of the same country drawn to different scales are similar figures. In order that polygons may be similar, for every angle of the one there must be a corresponding equal angle of the other and the sides about the equal angles must be proportional. J 625. 1. Triangles ABC, DEF, and GHJ are similar. How do the sides of ABC compare in length with the corresponding sides of DEF? of GHJ? F 2. The sides of the first two triangles are in the proportion of 1 to 2, and their areas are in the proportion of 1 to 4 (the squares of 1 and 2). The sides of the first and third triangles. are in the proportion of 1 to 3, and their areas in the proportion of 1 to 9. 3. Show in the same way that the sides of these squares are in the proportion of 1:23, and that their areas are in the proportion of 1:49, the squares of the sides. 626. 1. The corresponding sides or like dimensions of similar plane figures are proportional. 2. The areas of similar plane figures are proportional to the squares of their corresponding lines. 627. 1. If a rectangle is 4 inches long and 3 inches wide, find the width of a similar rectangle that is 8 inches long. 2. The area of a circle is 5 square inches. Find the area of a circle whose diameter is twice the diameter of the first. 3. The side of a square is 5 inches. Find the side of another square that contains 4 times as much area. 4. The sides of two regular octagons are as 1 to 3. What is the ratio of their areas? 5. A schoolroom has two square blackboards whose sides are 3 feet and 6 feet, respectively. What is the area of the first? Find the area of the second by applying the principle of similar figures. 6. A lady has two circular flower beds, one having a radius of 4 feet and the other a radius of 16 feet. How do they compare in area? 7. The sides of a triangle are 1 centimeter, 2 centimeters, and 3 centimeters. What are the sides of a similar triangle containing 25 times the area of the first? 8. If the ratio of two similar triangles is 16, what is the ratio of their bases? 9. By the principle of similar figures, find the height of a poplar tree that casts a 25-foot shadow when a boy 5 feet tall casts a shadow 6 feet long. -64 ft 10. When a telephone pole 30 feet high casts a shadow of 60 feet, what is the height of a church steeple that casts a shadow 300 feet long? 25 ft. |