Basic Mathematics & Logarithm
AM-HM Inequality
Grade 11
Question:
<p>Four pyramids have square bases with areas <i>A</i><sub>1</sub>, <i>A</i><sub>2</sub>, <i>A</i><sub>3</sub>, <i>A</i><sub>4</sub> and heights <i>h</i><sub>1</sub>, <i>h</i><sub>2</sub>, <i>h</i><sub>3</sub>, <i>h</i><sub>4</sub> respectively, all having the same volume <i>V</i>. Find the minimum value of \((A_1+A_2+A_3+A_4)(h_1+h_2+h_3+h_4)\).</p>
Step-by-Step Solution
Key Concept: Since all pyramids have equal volume V = (1/3)A·h, we have A·h = 3V (constant). Apply Cauchy-Schwarz inequality to the product of two sums where paired terms have a fixed product.
<p><strong>Step 1: Apply Volume Constraint</strong></p><p>For a square-based pyramid: V = (1/3)A·h</p><p>Since all four pyramids have the same volume V:</p><p>A₁h₁ = A₂h₂ = A₃h₃ = A₄h₄ = 3V</p><p><strong>Step 2: Apply Cauchy-Schwarz Inequality</strong></p><p>By Cauchy-Schwarz: (a₁² + a₂² + a₃² + a₄²)(b₁² + b₂² + b₃² + b₄²) ≥ (a₁b₁ + a₂b₂ + a₃b₃ + a₄b₄)²</p><p>Let aᵢ = √(Aᵢ) and bᵢ = √(hᵢ):</p><p>(A₁ + A₂ + A₃ + A₄)(h₁ + h₂ + h₃ + h₄) ≥ (√(A₁h₁) + √(A₂h₂) + √(A₃h₃) + √(A₄h₄))²</p><p><strong>Step 3: Substitute Constraints</strong></p><p>Since Aᵢhᵢ = 3V for each i:</p><p>(A₁ + A₂ + A₃ + A₄)(h₁ + h₂ + h₃ + h₄) ≥ (√(3V) + √(3V) + √(3V) + √(3V))²</p><p>= (4√(3V))² = 16(3V) = 48V</p><p><strong>Step 4: Equality Condition</strong></p><p>Equality in Cauchy-Schwarz occurs when A₁ = A₂ = A₃ = A₄ and h₁ = h₂ = h₃ = h₄</p><p>With constraint A·h = 3V: minimum = 48V</p><p>For V = 1/6: minimum value = 48(1/6) = <strong>8</strong></p>
Correct Answer: 8