# Cylinder of maximum volume and maximum lateral area inscribed in a cone

**Situation**

A right circular cylinder of radius r and height h is inscribed in a right circular cone of radius 6 m and height 12 m.

Part 1: Determine the radius of the cylinder such that its volume is a maximum.

A. 2 m

B. 4 m

C. 3 m

D. 5 m

Part 2: Determine the maximum volume of the cylinder.

A. 145.72 m^{3}

B. 321.12 m^{3}

C. 225.31 m^{3}

D. 201.06 m^{3}

Part 3: Determine the height of the cylinder such that its lateral area is a maximum.

A. 10 m

B. 8 m

C. 6 m

D. 4 m

**Solution**

$12 - h = 2r$

$ h = 12 - 2r$

For maximum volume of cylinder:

$V = \pi r^2(12 - 2r)$

$V = 2\pi(6r^2 - r^3)$

$\dfrac{dV}{dr} = 2\pi(12r -3r^2) = 0$

$12r -3r^2 = 0$

$r = 4 \, \text{ m}$ Part 1: [ B ]

$h = 12 - 2(4)$

$h = 4 \, \text{ m}$

$V_{max} = \pi(4^2)(4)$

$V_{max} = 201.062 \, \text{ m}^3$ Part 2: [ D ]

For maximum lateral surface area:

$A_L = 2\pi r(12 – 2r)$

$A_L = 4\pi(6r – r^2)$

$\dfrac{dA_L}{dr} = 4\pi(6 - 2r) = 0$

$6 - 2r = 0$

$r = 3 \, \text{ m}$

$h = 12 - 2(3)$

$h = 6 \, \text{ m}$ Part 3: [ C ]