The original question includes a diagram: A circle of radius 5 cm with its centre marked, and a rectangle inscribed in it with all four vertices on the circle: the cross section of the sphere and cylinder through their common centre.
A right circular cylinder is contained within a sphere of radius 5 cm in such a way that the whole of the circumferences of both ends of the cylinder are in contact with the sphere.
The diagram shows a planar cross section through the centre of the sphere and cylinder: a circle of radius 5 cm with an inscribed rectangle whose four corners lie on the circle.
Find, in cubic centimetres, the maximum possible volume of the cylinder.
- A250π
- B500π
- C1000π
- D250√33π
- E500√39π
- F1000√39π
Show the answer and worked solution
answer · E
- A250π
- B500π
- C1000π
- D250√33π
- E500√39π
- F1000√39π
Let the cylinder have radius r and half-height h, so its axis, the radius and a sphere radius form a right-angled triangle: r2 + h2 = 25. The volume is V = π r2 (2h) = 2π h(25 − h2), a function of the single variable h. Then dVdh = 2π(25 − 3h2) = 0 gives h = 5√3, which is a maximum since V vanishes at h = 0 and h = 5. Substituting back, V = 2π⋅5√3⋅503 = 500π3√3 = 500√39π.