The right circular cylinder of given surface and max. volume is such that its height is equal to radius of the base.

Answer

To solve the problem of finding the right circular cylinder of given surface area that maximizes volume with the condition that its height is equal to the radius of the base, we will follow these steps:

### Step 1: Define the variables
Let:
- \( r \) = radius of the base of the cylinder
- \( h \) = height of the cylinder
- \( S \) = total surface area of the cylinder

### Step 2: Write the formula for the surface area
The total surface area \( S \) of a right circular cylinder is given by:
\[ S = 2\pi r^2 + 2\pi rh \]
Here, \( 2\pi r^2 \) is the area of the two circular bases, and \( 2\pi rh \) is the lateral surface area.

### Step 3: Express height in terms of radius and surface area
Given that \( h = r \), we can substitute \( h \) in the surface area equation:
\[ S = 2\pi r^2 + 2\pi r(r) = 2\pi r^2 + 2\pi r^2 = 4\pi r^2 \]
Thus, we can express \( r \) in terms of \( S \):
\[ r^2 = \frac{S}{4\pi} \implies r = \sqrt{\frac{S}{4\pi}} \]

### Step 4: Write the formula for the volume
The volume \( V \) of the cylinder is given by:
\[ V = \pi r^2 h \]
Substituting \( h = r \):
\[ V = \pi r^2 r = \pi r^3 \]

### Step 5: Substitute \( r \) into the volume formula
Now substituting \( r = \sqrt{\frac{S}{4\pi}} \) into the volume formula:
\[ V = \pi \left(\sqrt{\frac{S}{4\pi}}\right)^3 = \pi \cdot \frac{S^{3/2}}{(4\pi)^{3/2}} = \frac{S^{3/2}}{4\sqrt{4\pi}} \]

### Step 6: Differentiate to find maximum volume
To maximize the volume, we can differentiate \( V \) with respect to \( r \) and set the derivative equal to zero. However, since we have already expressed \( V \) in terms of \( S \), we can see that the volume is maximized when \( h = r \).

### Step 7: Verify the condition
From our earlier calculations, we see that:
- The condition \( h = r \) holds true under the given constraints, confirming that the maximum volume occurs when the height is equal to the radius.

### Conclusion
Thus, the right circular cylinder of given surface area that maximizes volume indeed has its height equal to the radius of the base.

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