cylinder volume and surface area relation

soo the question is: there is a volume of 750cm^3 of molten metal available, find the minimum dimensions to create a cylinder.

it sounds quite vague from looking at it. ive tried using both the volume and total surface area formulas for a cylinder which are:

V=πr²h and TSA=2πr²+2πrh

soo substituting 750 as V you would get: 750=πr²h and rearranging that you could get either: h=750/πr² or r=√(750/πh) and substitute those to the TSA formula.

From there i was able to differentiate both functions i got from substituing but couldnt find any maximum or minimum values as i kept getting hyperbola functions and what not.

can someone please find r and h? thanks in advance (Wink)

Re: cylinder volume and surface area relation

What is the meaning of "minimum dimensions to create a cylinder" ?

There are two dimensions : diameter=2r and height=h.

If you suppose that only one is minimum, the other will be maximum, which probably isn't what is expected.

So, it is logical to think that we are looking for both equal and minimum : h=2r The result is easy to obtain in this case.

A more complicated hypothesis might be to consider the largest distance between two points of the cylinder : L=sqrt(h²+(2r)²) and two find the dimensions so that L be minimum.

Since r=√(750/πh), we have L² = h²+4*750/nh from which a minimum valune for L can be derived.