Wentworth's Plane and Solid Geometry · Book VII, Exs. 643–656 · Book 8: Book VII: Polyhedrons, Cylinders, and Cones
98. Problems of computation: prisms
Goal: Your student computes surfaces and volumes of cubes, boxes and prisms, including real-life tanks and cisterns.
You'll need
- The book, open to these pages
- A ruler (straightedge) and a pair of compasses
- Sharp pencil, eraser and plain paper (or a geometry notebook)
- A calculator (optional)
What's in the book
Page 306: Problems of Computation, Ex. 643–656.
The lesson, step by step
Warm up
Ask: Volume of a prism? Lateral area of a right prism? Area of a triangle from its three sides?
Boxes and cubes
Ex. 643. If the edge of a cube is 15 inches, find the area of the total surface of the cube.
Ex. 644. If the length of a rectangular parallelopiped is 10 inches, its width 8 inches, and its height 6 inches, find the area of its total surface.
Prisms
Ex. 645. Find the volume of a right triangular prism, if its height is 14 inches, and the sides of the base are 6, 5, and 5 inches.
Ex. 646. The base of a right prism is a rhombus, one side of which is 10 inches, and the shorter diagonal is 12 inches. The height of the prism is 15 inches. Find the entire surface and the volume.
Cisterns
Ex. 649. An open cistern 6 feet long and 4½ feet wide holds 108 cubic feet of water. How many square feet of lead will it take to line the sides and bottom?
First find the depth. Then add the bottom and the four sides.
Letters instead of numbers
Your student does Ex. 653 and 655, giving answers as formulas.
Wrap up
Check the answers. Your student explains one solution aloud.
Tip
The book prints no answers. Where a problem needs a fact the book does not state (like the weight of water), say so and agree on a value: a cubic foot of water weighs about 62.5 pounds.
Answers to the book's problems
Ex. 643: 1350 sq in. Ex. 644: 376 sq in. Ex. 645: base 12 sq in, volume 168 cu in. Ex. 646: the longer diagonal is 16 in, base 96 sq in; surface 600 + 192 = 792 sq in; volume 1440 cu in. Ex. 647: base 25√3, about 43.3 sq in; volume about 5196 cu in, or about 3.01 cu ft. Ex. 648: depth 3½ ft; lead 12 + 50 1/6 = 62 1/6 sq ft. Ex. 649: depth 4 ft; 27 + 84 = 111 sq ft. Ex. 650: if the iron is 2 in thick on the four sides and the bottom (open top), the iron is about 9.21 cu ft; at 62.5 lb per cubic foot of water it weighs about 4146 lb empty and about 6255 lb full. Ex. 651: about 31,177 cu in, or about 18.04 cu ft. Ex. 652: 2 tons = 4000 lb = 64 cu ft of water (at 62.5 lb per cu ft), so the edge is 4 ft. Ex. 653: 6a², a³, a√3. Ex. 654: the edge is a/√2, the volume a³√2/4. Ex. 655: 2(ab + bc + ca), abc, √(a² + b² + c²). Ex. 656: the dimensions are mk, nk, pk where k = ∛(V/mnp).
Worked out for this site from the scan. If a number in your copy differs, trust the book.