Let w and l be the dimensions of the paper. We know that 2(l+w)=100 or l+w=50.
Volume of a cylinder is given by:
h=l; then r=w/(2*pi) and V=w2l/(4*pi). Using l+w=50 we obtain:
V'(w)=(100w-3w2)/(4*pi); V'(w)=0 at w=0 and w=100/3. Obviously, w=100/3 cm gives maximum volume. l=50-100/3=50/3 cm.
Second case h=w, then the same considerations are applied and the same result is obtained for volume.
Answer: 100/3 cm and 50/3 cm; Vmax=125000/(27*pi)