<p>If \(x = 4t^3\) and \(y = 3t^4\) (parametric form), and \(\dfrac{d^2x/dy^2}{(dx/dy)^n}\) = constant, find <em>n</em>. Then compute the sum \(\dfrac{4/5}{1 - 1/n}\).</p>
Step-by-Step Solution
Key Concept: Convert parametric derivatives using chain rule: dx/dy = (dx/dt)/(dy/dt), then find d²y/dx² by differentiating dx/dy with respect to t and dividing by dx/dt. Match the power of dx/dy to find n.
<p><strong>Step 1: Find first derivatives</strong></p><p>Given: x = 4t³, y = 3t⁴</p><p>dx/dt = 12t², dy/dt = 12t³</p><p><strong>Step 2: Find dx/dy</strong></p><p>dx/dy = (dx/dt)/(dy/dt) = 12t²/(12t³) = 1/t</p><p><strong>Step 3: Find d²x/dy²</strong></p><p>d²x/dy² = d/dy(dx/dy) = d/dt(1/t) · dt/dy = (-1/t²) · (1/(12t³)) = -1/(12t⁵)</p><p><strong>Step 4: Express in terms of dx/dy</strong></p><p>Since dx/dy = 1/t, we have t = 1/(dx/dy), so t⁵ = 1/(dx/dy)⁵</p><p>d²x/dy² = -12(dx/dy)⁵</p><p><strong>Step 5: Identify the power n</strong></p><p>From d²x/dy²/(dx/dy)ⁿ = constant, we get:</p><p>-12(dx/dy)⁵/(dx/dy)ⁿ = constant</p><p>This requires n = 5</p><p><strong>Step 6: Calculate the final answer</strong></p><p>(4/5)/(1 - 1/5) = (4/5)/(4/5) = <strong>1</strong></p>
Correct Answer: 1