Wentworth's Plane and Solid Geometry · Book VII, Exs. 702–711 · Book 8: Book VII: Polyhedrons, Cylinders, and Cones
115. Frustums of pyramids and of cones
Goal: Your student computes areas and volumes of frustums in practical problems.
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 357: Exercises on frustums of pyramids and of cones, Ex. 702–711.
The lesson, step by step
Warm up
Ask: Lateral area and volume of a frustum of a cone? of a pyramid?
A funnel and a column
Ex. 702. How many square feet of tin are required to make a funnel, if the diameters of the top and bottom are 28 inches and 14 inches, respectively, and the height is 24 inches?
Ex. 703. Find the expense, at 60 cents a square foot, of polishing the curved surface of a marble column in the shape of the frustum of a right circular cone whose slant height is 12 feet, and the radii of whose bases are 3 feet 6 inches and 2 feet 4 inches, respectively.
Volumes
Ex. 704. The slant height of the frustum of a regular pyramid is 20 feet; the sides of its square bases 40 feet and 16 feet. Find the volume.
Ex. 709. A Dutch windmill in the shape of the frustum of a right cone is 12 meters high. The outer diameters at the bottom and the top are 16 meters and 12 meters, the inner diameters 12 meters and 10 meters. How many cubic meters of stone were required to build it?
A chimney
Ex. 710. The chimney of a factory has the shape of a frustum of a regular pyramid. Its height is 180 feet, and its upper and lower bases are squares whose sides are 10 feet and 16 feet, respectively. The flue is throughout a square whose side is 7 feet. How many cubic feet of material does the chimney contain?
Wrap up
Check the answers together. Ex. 706 makes a good challenge for another day.
Answers to the book's problems
Ex. 702: radii 14 and 7 in, slant height 25 in; 525π ≈ 1649 sq in ≈ 11.45 sq ft. Ex. 703: about 219.9 sq ft; about $131.95. Ex. 704: altitude 16 ft; 13,312 cu ft. Ex. 705: about 1.98 ft. Ex. 706: R² + r² + Rr = 63 (with π = 22/7) and R + r = 9, so R = 6 ft, r = 3 ft. Ex. 707: lateral area 144 sq ft; volume about 78.8 cu ft. Ex. 708: the difference is ⅓ of the altitude (in cubic feet, with the altitude in feet). Ex. 709: 228π ≈ 716.3 cu m. Ex. 710: 30,960 − 8820 = 22,140 cu ft. Ex. 711: R − r = √(L² − H²) and R + r = S/(πL); then V = ⅓πH(R² + r² + Rr).
Worked out for this site from the scan. If a number in your copy differs, trust the book.