Applications of Derivatives
Related Rates
Grade 12
Question:
<p>A rod of length 5 has ends A and B sliding along the curve \(y = 2x^2\). Let \(x_A\) and \(x_B\) be the x-coordinates of the ends. When A is at \((1, 2)\) and B is at \((0, 0)\), find \(\frac{dx_B}{dx_A}\).</p>
<p>(a) 16</p>
<p>(b) 8</p>
<p>(c) 9</p>
<p>(d) 2</p>
Step-by-Step Solution
Key Concept: Apply the same implicit differentiation technique with different initial conditions to find the derivative at a different position on the curve.
<p><strong>Solution:</strong> Using the same constraint equation and differentiation approach as Ex. 21:</p><p>$$2(x_B - x_A)(D - 1) + 8(x_B - x_A)(2x_B D - 2x_A) = 0$$</p><p>When $x_A = 1$ and $x_B = 0$:</p><p>$$2(-1)(D - 1) + 8(-1)(0 - 2) = 0$$</p><p>$$-2(D - 1) + 16 = 0$$</p><p>$$-2D + 2 + 16 = 0$$</p><p>$$2D = 18$$</p><p>$$D = 9$$</p><p>∴ Answer is (c).</p>
Correct Answer: C