Question:

<p><strong>Step 1:</strong> Apply the Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B}$</p><p>This gives us: $\frac{5}{\sin A} = \frac{4}{\sin B}$, so $\sin A = \frac{5\sin B}{4}$</p><p><strong>Step 2:</strong> Use the given condition $\cos(A - B) = \frac{31}{32}$</p><p>Expand: $\cos A \cos B + \sin A \sin B = \frac{31}{32}$</p><p><strong>Step 3:</strong> From the Law of Cosines, express $\cos A$ and $\cos B$:</p><p>$\cos A = \frac{b^2 + c^2 - a^2}{2bc} = \frac{16 + c^2 - 25}{2 \cdot 4 \cdot c} = \frac{c^2 - 9}{8c}$</p><p>$\cos B = \frac{a^2 + c^2 - b^2}{2ac} = \frac{25 + c^2 - 16}{2 \cdot 5 \cdot c} = \frac{c^2 + 9}{10c}$</p><p><strong>Step 4:</strong> From Law of Sines: $\sin A = \frac{5\sin B}{4}$. Using $\sin^2 + \cos^2 = 1$:</p><p>$\sin B = \sqrt{1 - \cos^2 B} = \sqrt{1 - \left(\frac{c^2 + 9}{10c}\right)^2}$</p><p><strong>Step 5:</strong> Substitute into $\cos A \cos B + \sin A \sin B = \frac{31}{32}$ and simplify. After algebraic manipulation using the constraint that $\sin A = \frac{5\sin B}{4}$:</p><p>$\frac{c^2 - 9}{8c} \cdot \frac{c^2 + 9}{10c} + \frac{5\sin B}{4} \cdot \sin B = \frac{31}{32}$</p><p><strong>Step 6:</strong> This simplifies to finding c. Testing $c = 6$:</p><p>$\cos A = \frac{36 - 9}{48} = \frac{27}{48} = \frac{9}{16}$</p><p>$\cos B = \frac{36 + 9}{60} = \frac{45}{60} = \frac{3}{4}$</p><p>$\sin A = \sqrt{1 - \frac{81}{256}} = \frac{\sqrt{175}}{16} = \frac{5\sqrt{7}}{16}$</p><p>$\sin B = \sqrt{1 - \frac{9}{16}} = \frac{\sqrt{7}}{4}$</p><p>$\cos(A-B) = \frac{9}{16} \cdot \frac{3}{4} + \frac{5\sqrt{7}}{16} \cdot \frac{\sqrt{7}}{4} = \frac{27}{64} + \frac{35}{64} = \frac{62}{64} = \frac{31}{32}$ ✓</p><p><strong>∴ Answer:</strong> B</p>