The ball chapter and stereometry review
Answers are for parents and teachers.
1
The sphere (x − 1)² + (y + 2)² + (z − 3)² = 36 is given. Is the origin inside the ball, on the sphere or outside?
1 + 4 + 9 = 14 < 36: the origin is inside the ball.
2
Find the volume of the space between a ball of radius 4 and its circumscribed cylinder.
V_cylinder = π · 16 · 8 = 128π, V_ball = 256π/3; the difference is 128π − 256π/3 = 128π/3.
3
A cone of height 4 stands on a cylinder (r = 3, height 5) with coinciding bases. Find the volume and surface of the solid.
V = π · 9 · 5 + ⅓ π · 9 · 4 = 45π + 12π = 57π. The cone’s slant height is 5: S = π · 9 (bottom base) + 2π · 3 · 5 (lateral) + π · 3 · 5 (cone) = 9π + 30π + 15π = 54π.
4
For A (1, 0, 2) and B (3, 4, 6) find the midpoint and the length of the segment.
The midpoint is (2, 2, 4); AB = √(4 + 16 + 16) = 6.
5
Why does no convex polyhedron with 7 edges exist?
Each vertex has at least 3 edges and each face at least 3 edges: 3V ≤ 2E and 3F ≤ 2E. For E = 7 this gives V ≤ 4 and F ≤ 4, so V + F ≤ 8, but Euler’s formula demands V + F = E + 2 = 9. A contradiction.