Surface area:volume ratio increases with more extreme cylinder shapes for fixed volume

Steps:

• Recall SA/V = 2/r + 2/h for a cylinder.
• Assume equal volume, so h ∝ 1/r²; ratio = 2/r + 2r² (normalized V=π).
• Z has balanced r and h (e.g., h≈r), minimizing ratio via calculus (derivative zero at optimal r).
• X is tall/thin (small r, large h), giving moderate ratio; Y is short/fat (large r, small h), giving largest ratio.

Why C is correct:

• C orders from smallest to largest ratio (Z balanced < X tall/thin < Y short/fat), matching formula where extremes increase SA/V relative to balanced shape.

Why the others are wrong:

• A reverses order, ignoring balanced minimum.
• B swaps X and Z, underestimating thin cylinder's ratio.
• D starts with medium Y, missing Z's lowest ratio.

Final answer: C