<p>Given \(y - x^{3/2} = 7\). Find the shortest distance from the point \(A\left(\dfrac{1}{2}, 7\right)\) to the curve.</p>
Step-by-Step Solution
Key Concept: The shortest distance from a point to a curve occurs along the normal to the curve. At the shortest distance point, the line joining the point to the curve is perpendicular to the tangent, meaning the slope of this joining line equals the negative reciprocal of dy/dx.
<p><strong>Step 1:</strong> From the curve equation: y = x^(3/2) + 7</p><p><strong>Step 2:</strong> Find dy/dx = (3/2)x^(1/2) = (3/2)√x</p><p><strong>Step 3:</strong> For shortest distance, the line from point A(1/2, 7) to point P(x, x^(3/2) + 7) on the curve must be normal to the curve.</p><p><strong>Step 4:</strong> Slope from A to P: m = (x^(3/2) + 7 - 7)/(x - 1/2) = x^(3/2)/(x - 1/2)</p><p><strong>Step 5:</strong> For normality: m · (dy/dx) = -1</p><p>x^(3/2)/(x - 1/2) · (3/2)√x = -1</p><p><strong>Step 6:</strong> Simplifying: (3/2)x²/(x - 1/2) = -1</p><p>3x² = -2(x - 1/2) = -2x + 1</p><p>3x² + 2x - 1 = 0</p><p>(3x - 1)(x + 1) = 0</p><p><strong>Step 7:</strong> Since x ≥ 0 for x^(3/2), we get x = 1/3</p><p><strong>Step 8:</strong> Point P: (1/3, (1/3)^(3/2) + 7) = (1/3, 1/(3√3) + 7)</p><p><strong>Step 9:</strong> Distance = √[(1/3 - 1/2)² + (1/(3√3))²] = √[1/36 + 1/27] = √[(3 + 4)/108] = √(7/108) ≈ 0.2546</p><p>∴ Answer: 0.2546</p>
Correct Answer: 0.2546