Wentworth's Plane and Solid Geometry · Book VIII, Exs. 791–810 · Book 9: Book VIII: The Sphere
130. Numerical exercises: volumes
Goal: Your student computes volumes of spheres, shells, wedges, sectors, segments and spherical pyramids.
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
Pages 403–404: Numerical Exercises, Ex. 791–810.
The lesson, step by step
Warm up
Ask: Volume of a sphere? of a spherical sector? of a segment of one base?
Spheres
Ex. 791. Find the volume of a sphere, if the diameter is (i) 13 inches; (ii) 3 feet 6 inches; (iii) 10 feet 6 inches; (iv) 14.7 feet.
Ex. 798. An iron ball 4 inches in diameter weighs 9 pounds. Find the weight of an iron shell 2 inches thick, whose external diameter is 20 inches.
Wedges and pyramids
Ex. 799. The radius of a sphere is 7 feet. What is the volume of a wedge whose angle is 36°?
Ex. 806. Find the volume of a triangular spherical pyramid, if the angles of the spherical triangle which forms its base are each 100°, and the radius of the sphere is 7 feet.
Segments
Ex. 803. The inside of a wash-basin is in the shape of the segment of a sphere; the distance across the top is 16 inches, and its greatest depth is 6 inches. Find how many pints of water it will hold, reckoning 7½ gallons to the cubic foot.
Formulas
Your student tries Ex. 794 or 795.
Wrap up
Check the answers together.
Answers to the book's problems
Ex. 791: (i) about 1150 cu in; (ii) about 22.45 cu ft; (iii) about 606.1 cu ft; (iv) about 1663 cu ft. Ex. 792: (i) about 5.25 ft; (ii) about 7 ft; (iii) about 12.6 ft; (iv) about 4.2 yd. Ex. 793: about 1539 cu ft. Ex. 794: V = C³/(6π²). Ex. 795: R = ∛(3V/4π). Ex. 796: R = √1.5 ≈ 1.22. Ex. 797: (4/3)πR³ : 8R³ = π : 6. Ex. 798: the shell is (8000 − 4096)/64 times the ball; 549 lb. Ex. 799: one tenth of the sphere; about 143.7 cu ft. Ex. 800: 60°. Ex. 801: 1 cu ft. Ex. 802: the plane is 12 in from the centre; the smaller segment (height 8 in) is about 3485 cu in; the larger (height 32 in) about 30,025 cu in. Ex. 803: 228π ≈ 716.3 cu in, about 24.9 pints. Ex. 804: H = S/(2πR) with R = ∛(3V/4π). Ex. 805: 154.5π ≈ 485.4 cu ft. Ex. 806: E = 120; about 239.5 cu ft. Ex. 807: R = 14, E = 145; about 2315 cu ft. Ex. 808: E = 90; about 22.4 cu ft. Ex. 810: the floating part is a segment whose zone is ¾ of the surface (height 3R/2); its volume is 9πR³/8, so the weight is 9πR³s/8 (in units of the weight of a unit volume of water).
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