Consider a cuboid of sides , and and a closed hemisphere of radius . If the sum of their surface areas is constant , then the ratio , for which the sum of their volumes is maximum, is

Question

Consider a cuboid of sides ,  and  and a closed hemisphere of radius . If the sum of their surface areas is constant , then the ratio , for which the sum of their volumes is maximum, is

Canvas Classes · Accepted Answer

🧠 Two Surfaces, One Constraint — Use Lagrange Multipliers (or Substitution)

Read the question. Cuboid with dimensions . Hemisphere of radius . Sum of surface areas  (constant). Maximize sum of volumes.

Surface areas:
- Cuboid: .
- Closed hemisphere (curved + flat disk): .

Constraint: .

Volumes:
- Cuboid: .
- Hemisphere: .

Maximize: .

First move: use Lagrange multipliers.

🗺️ Working It Out

Lagrange: .

, .

Two equations:
.
.

Set equal: , giving .

So , i.e., .

⚡ The Smart Move

More questions from this chapter

More questions from this chapter