The height of the cylinder of max. volume that can be inscribed in a sphere of radius a is

Options

A. `(2a)/sqrt3`
B. `a/sqrt2`
C. `(5a)/4`
D. None

Answer

To find the height of the cylinder of maximum volume that can be inscribed in a sphere of radius \( a \), we can follow these steps:

### Step 1: Understand the Geometry
We have a sphere of radius \( a \) and a cylinder inscribed within it. Let the radius of the cylinder be \( r \) and the height of the cylinder be \( h \).

### Step 2: Relate the Cylinder and Sphere
The cylinder is inscribed in the sphere, which means that the diagonal of the cylinder (from the center of the base to the top edge) will equal the radius of the sphere. If we let the height of the cylinder be \( h \) and the radius of the cylinder be \( r \), we can use the Pythagorean theorem in the triangle formed by the radius of the sphere, the radius of the cylinder, and half the height of the cylinder:
\[ a^2 = r^2 + \left(\frac{h}{2}\right)^2 \]

### Step 3: Express Volume of the Cylinder
The volume \( V \) of the cylinder can be expressed as:
\[ V = \pi r^2 h \]

### Step 4: Substitute for \( r \)
From the Pythagorean theorem, we can express \( r^2 \) in terms of \( h \):
\[ r^2 = a^2 - \left(\frac{h}{2}\right)^2 \]
Substituting this into the volume formula gives:
\[ V = \pi (a^2 - \left(\frac{h}{2}\right)^2) h \]

### Step 5: Simplify the Volume Expression
Expanding this, we have:
\[ V = \pi \left(a^2h - \frac{h^3}{4}\right) \]

### Step 6: Differentiate the Volume
To find the maximum volume, we differentiate \( V \) with respect to \( h \):
\[ \frac{dV}{dh} = \pi \left(a^2 - \frac{3h^2}{4}\right) \]

### Step 7: Set the Derivative to Zero
Setting the derivative equal to zero to find critical points:
\[ a^2 - \frac{3h^2}{4} = 0 \]
Solving for \( h \):
\[ \frac{3h^2}{4} = a^2 \implies h^2 = \frac{4a^2}{3} \implies h = \frac{2a}{\sqrt{3}} \]

### Step 8: Conclusion
Thus, the height of the cylinder of maximum volume that can be inscribed in a sphere of radius \( a \) is:
\[ h = \frac{2a}{\sqrt{3}} \]