A cylinder inscribed in a cone

A circular cylinder is inscribed in a right circular cone of base radius 2c2c and vertical height cc. One of the circular ends of the cylinder lies on the base of the cone and the other end is in contact with the inner surface of the cone. If the cylinder has base radius xx, find expressions for the volume VV and the total surface area SS of the cylinder in terms of cc and xx. If cc is fixed and xx varies, find the maximum value of VV. marks

Prove that SS cannot exceed 8πc28\pi c^2. marks

[2010/NYJC/Prelim/II/Q2]

Total marks: 10

Solution

Volume

Similar triangles

A cylinder of radius xx is inscribed in a cone of base radius 2c2c and height cc, with one end on the base. Find the volume VV of the cylinder in terms of cc and xx.

Hint

  1. Let the cylinder have height hh.

  2. The top rim of the cylinder touches the cone's surface: use similar triangles to relate hh and xx.

Answer · Volume

V=π2x2(2c−x)V=\frac\pi2x^2(2c-x).

Surface area

Surface areas of solids

Find the total surface area SS of the cylinder in terms of cc and xx.

Hint

  1. Two circular ends plus the curved surface.

Answer · Surface area

S=πx(x+2c)S=\pi x(x+2c).

Maximum volume

Optimisation

If cc is fixed and xx varies, find the maximum value of VV.

Explore this idea

Hint

  1. Differentiate VV with respect to xx and set it to 0.

  2. Verify it is a maximum.

  3. Substitute back for Vmax⁡V_{\max}.

Answer · Maximum volume

Vmax⁡=16πc3/27V_{\max}=16\pi c^3/27 at x=4c/3x=4c/3.

Explore the idea

Volume against cylinder radius (c = 1)

The volume rises, peaks at x = 4/3, then falls to 0 as the cylinder becomes as wide as the cone.

Position along the curvex = 0.5
Try an example
Test a critical position

Buttons use full-precision reviewed positions; displayed coordinates are rounded.

f(x)
0.58905
Gradient f′(x)
1.9635
positive

For 0<x<4/3 the gradient is positive: the volume increases.

Selected values are within the original drawing windows.

  • V(x) for c = 1
  • Tangent
  • DV/dx
  • Join P to Q; the dashed legs show Δx and Δy
Key idea

VV is zero at both ends of 0≤x≤2c0\le x\le2c, so a maximum lies between; the gradient test finds it at x=4c3x=\frac{4c}3.

Ideas to try
Is the maximum at the half-way value x=cx=c?
Surface-area bound

Bounding a function

Prove that SS cannot exceed 8πc28\pi c^2.

Explore this idea

Hint

  1. S=πx(x+2c)S=\pi x(x+2c) is increasing for x>0x>0.

  2. The cylinder's radius is at most 2c2c (the cone's base radius).

Answer · Surface-area bound

S<8πc2S<8\pi c^2, hence S≤8πc2S\le8\pi c^2.

c>0,0<x<2cc>0,0<x<2c; equality only in the zero-height limiting cylinder.

Explore the idea

S rises all the way to 8πc² and never exceeds it

Shown for c = 1: S(x) = πx(x + 2) for 0 < x < 2.

Switch on a feature to add it to the sketch and see how it is found. A full-mark sketch shows all of them.

  • S(x)
  • Bound 8πc²
Ideas to try
Why is there no maximum of SS itself?