<p>The length of three unequal edges of a rectangular solid block are in GP. The volume of the block is 216 cm³ and the total surface area is 252 cm². The length of the longest edge is</p>
Step-by-Step Solution
Key Concept: Set the three unequal edges in GP as a/r, a, ar (where r is the common ratio). Use the volume and surface area conditions to form equations that determine a and r, then identify the longest edge.
<p><strong>Step 1:</strong> Let the three unequal edges be a/r, a, and ar where a > 0 and r > 0 (r ≠ 1 since edges are unequal).</p><p><strong>Step 2:</strong> Use the volume condition:<br/>Volume = (a/r) × a × ar = a³ = 216<br/>Therefore, a = 6 cm</p><p><strong>Step 3:</strong> Use the surface area condition for a rectangular solid:<br/>Total surface area = 2(a/r · a + a · ar + ar · a/r)<br/>252 = 2(a²/r + a²r + a²)<br/>126 = a²(1/r + r + 1)<br/>126 = 36(1/r + r + 1)<br/>126/36 = 1/r + r + 1<br/>3.5 = 1/r + r + 1<br/>2.5 = 1/r + r</p><p><strong>Step 4:</strong> Multiply by r:<br/>2.5r = 1 + r²<br/>r² - 2.5r + 1 = 0<br/>r² - (5/2)r + 1 = 0<br/>2r² - 5r + 2 = 0<br/>(2r - 1)(r - 2) = 0<br/>r = 1/2 or r = 2</p><p><strong>Step 5:</strong> Since r ≠ 1 (edges are unequal), we have r = 2 or r = 1/2.<br/>If r = 2: edges are 6/2 = 3, 6, 6×2 = 12<br/>If r = 1/2: edges are 6/(1/2) = 12, 6, 6×(1/2) = 3<br/>Both give the same three edges: 3, 6, and 12 cm</p><p><strong>Step 6:</strong> Verify volume: 3 × 6 × 12 = 216 ✓<br/>Verify surface area: 2(3×6 + 6×12 + 12×3) = 2(18 + 72 + 36) = 2(126) = 252 ✓</p><p><strong>Step 7:</strong> The longest edge is 12 cm.</p><p><strong>∴ Answer:</strong> a</p>
Correct Answer: a