Tub A is a frustum with end radii 14 cm and 6 cm; Tub B is a frustum with end radii 11 cm and 9 cm. Both are 12 cm deep. Which holds more, and why?

Capacity depends on $R^2+r^2+Rr$, not on $R+r$. For A: 196 + 36 + 84 = 316; for B: 121 + 81 + 99 = 301. So A holds more: $\frac{1}{3}\times \frac{22}{7}\times 12\times 316=3972.57{\text{cm}}^3$ against $3784{\text{cm}}^3$. Equal sums $R+r$ do NOT give equal volumes, and curved surface area is a different quantity from capacity.