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°, 45° and 60° respectively. Let PQ = a and QR = b. What is PN equal to?
A.((3-√3)/2)a
B.((3+√3)/2)a✓ Correct
C.((3-√3)/4)a
D.((3+√3)/4)a
Explanation
Let MN = h. From P: tan30° = h/PN, so PN = h√3. From Q: tan45° = h/QN, so QN = h. From R: tan60° = h/RN, so RN = h/√3. PQ = PN - QN = h√3 - h = h(√3 - 1) = a, giving h = a/(√3-1) = a(√3+1)/2. Therefore PN = h√3 = a√3(√3+1)/2 = a(3+√3)/2.
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