The ball and the sphere
Answers are for parents and teachers.
1
A sphere with its centre on the Oz axis passes through (0, 0, 0) and (0, 4, 2). Write its equation.
Centre (0, 0, c): c² = 16 + (2 − c)² ⇒ 4c = 20, c = 5, R = 5; x² + y² + (z − 5)² = 25.
2
A section of a ball of radius R has half the area of a great circle. How far is the cutting plane from the centre?
πr² = πR²/2 ⇒ r² = R²/2; d² = R² − r² = R²/2, so d = R√2/2.
3
A tangent is drawn to a ball of radius 12 from a point 20 away from the centre. Find the length of the tangent.
t = √(400 − 144) = 16.
4
A cube is inscribed in a ball of radius 6. Find the volume of the cube.
The diagonal is 12 = a√3, so a = 4√3; V = a³ = 64 · 3√3 = 192√3.
5
Why is the great circle the largest section of a ball?
The section radius r = √(R² − d²) is largest when d = 0, giving r = R. As d grows (moving away from the centre) r decreases.