A cylindrical gas container is closed at the top and open at the bottom. If the iron plate of the top is times as thick as the plate forming the cylindrical sides, the ratio of the radius to the height of the cylinder using minimum material for the same capacity is :
The explanation for the correct option:
Step 1: Find the surface area of the cylinder.
Let, represents the thickness of the cylindrical sides.
So, the thickness of the iron plate at the top
We know that the volume of the cylinder is:
And the surface area of the cylinder is:
if be the thickness of the sides then that of the top will be
So the surface area would be:
Step 2: Find the first derivative of the surface area of the cylinder:
Now, So that we get the value of in terms of .
Step 3: Find the required ratio:
when, or
Hence at this value surface area will be minimum
Therefore, option (C) is the correct answer.