Question

A right circular cylinder is inscribed in a sphere of radius r. Find the dimensions of such a cylinder with the largest possible volume (your answer may depend on r).

Answer 1

Answer:

Largest volume in terms of r =  4r^3π/ 3√3 or you could write it as  4√3r^3π/9.

Step-by-step explanation:

Answer 2

The height and radius of the right circular cylinder inscribed in sphere are $h = 0.716\cdot V_{c}^{1/3}$ and $r = \sqrt{R^{2}-0.128\cdot V_{c}^{1/3}}$, respectively.

Geometrically speaking, the volumes of the sphere ($V_{s}$) and the right circular cylinder ($V_{c}$) are defined by the following formulae:

Sphere

$V_{s} = \frac{4\pi}{3}\cdot R^{3}$ (1)

Cylinder

$V_{c} = \pi\cdot r^{2}\cdot h$ (2)

Where:

• $R$ - Radius of the sphere.
• $r$ - Radius of the cylinder.
• $h$ - Height of the cylinder.

In addition, we find the following relationship by the Pythagorean theorem:

$R = \sqrt{r^{2}+\frac{h^{2}}{4} }$ (3)

By (3) we clear $r^{2}$:

$r^{2} = R^{2}-\frac{h^{2}}{4}$

(3) in (2):

$V_{c} = \pi \cdot \left(R^{2}-\frac{h^{2}}{4} \right)\cdot h$

$V_{c} = \pi\cdot R^{2}\cdot h - \frac{\pi}{4}\cdot h^{3}$ (4)

By (1):

$R = \sqrt[3]{\frac{3\cdot V_{s}}{4\pi} }$

(1) in (4):

$V_{c} = \pi\cdot \left(\frac{3\cdot V_{s}}{4\pi} \right)^{2/3}\cdot h -\frac{\pi}{4}\cdot h^{3}$ (5)

Now we proceed to perform first and second derivative tests:

FDT

$\pi\cdot \left(\frac{3\cdot V_{c}}{4\pi} \right)^{2/3}-\frac{3\pi}{4}\cdot h^{2} =0$

$\left(\frac{3\cdot V_{c}}{4\pi} \right)^{2/3}= \frac{3}{4}\cdot h^{2}$

$0.385\cdot V_{c}^{2/3} = 0.75\cdot h^{2}$

$h = 0.716\cdot V_{c}^{1/3}$ (6)

SDT

$V''_{c} = -\frac{3\pi}{2}\cdot h$ (7)

Since $h > 0$ and $V_{c}'' < 0$, the critical value is associated to the maximum volume of the cylinder.

The dimensions of the right circular cylinder are the following:

By (6):

$h = 0.716\cdot V_{c}^{1/3}$

By (3):

$r = \sqrt{R^{2}-0.25\cdot (0.716\cdot V_{c}^{1/3})^{2}}$

$r = \sqrt{R^{2}-0.128\cdot V_{c}^{1/3}}$

The height and radius of the right circular cylinder inscribed in sphere are $h = 0.716\cdot V_{c}^{1/3}$ and $r = \sqrt{R^{2}-0.128\cdot V_{c}^{1/3}}$, respectively.

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