What is MN equal to?
Consider the following for the two (02) items that follow : The top (M) of a tower is observed from three points P, Q and R lying in a horizontal straight line which passes directly along the foot (N) of the tower. The angles of elevations of M from P, Q and R are 30^{\circ}, 45^{\circ} and 60^{\circ} respectively. Let PQ=a and QR=b.
- A. (\frac{3+\sqrt{3}}{2})b ✓
- B. (\frac{3-\sqrt{3}}{2})b
- C. (\frac{3-\sqrt{3}}{4})b
- D. (\frac{3+\sqrt{3}}{4})b
Correct Answer: A. (\frac{3+\sqrt{3}}{2})b
Explanation
Let MN = h. From QR = b, we have QR = QN - RN = h - \frac{h}{\sqrt{3}} = h(\frac{\sqrt{3}-1}{\sqrt{3}}) = b. Solving for h gives h = \frac{b\sqrt{3}}{\sqrt{3}-1} = \frac{b\sqrt{3}(\sqrt{3}+1)}{2} = b(\frac{3+\sqrt{3}}{2}).