Prove that any cylindrical can of volume k cubic units that is to be made using a minimum of material must have the height equal to the diameter.
material is equivalent to the surface area ...
solve the volume equation for in terms of and , then substitute the derived expression for into the surface area equation ...
find and determine the value of that minimizes ... finally determine from your derived expression.
also remember that is a constant.
So, the given condition is
= k (Eq. 1)
And we need to minimize the surface area given by
Step 1: Solve equation 1 for h in terms of r and substitue into equation 2.
Step 2: Differentiate equation 2 wrt r. Set derivative equal to zero and solve for r.
Step 3: Substitute r back into equation 1 to get h.
You should then see that r and h will be equal.