Optimization Application

Question

for a closed cylinder of radius r cm in height h cm find the dimensions given the minimum surface area of 6cm^3

Answer 1

The problem is not clearly formulated. Let's assume that you meant that the surface area is 6 cm² and we want to maximize the volume given that surface area. This would be a standard optimization problem.

The surface area of a cylinder is 2pi*r²(top+bottom surfaces) + 2pi*r*h (lateral surface) = 6, let's solve this for h, we get h = (6-pi*r²)/(pi*r).

Volume of the cylinder is pi*r²*h, plug in the expression for h in here, then simplify and take the first derivative.

You are looking for extrema, so you want to set the first derivative equal to zero, which gives us

6-3*pi*r²=0, which gives r=sqrt(2/pi), we ignore negative root. The second derivative is equal to -6*pi*r, which is negative, which means this is a maximum.