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°, 45° and 60° respectively. Let PQ = a and QR = b. What is PN equal to?
B. ( 3+ 3 2 )a
Let h be the height of the tower MN. In MNP, 30^ = h PN PN = h 3 In MNQ, 45^ = h QN QN = h In MNR, 60^ = h RN RN = h 3 Since the angles of elevation increase as we move closer to the tower, the points P, Q, R lie on the same side of N in the order P, Q, R, N. Given PQ = a, we have PN - QN = a. h 3 - h = a h( 3 - 1) = a h = a 3 - 1 Rationalizing the denominator, we get: h = a( 3 + 1) ( 3 - 1)( 3 + 1) = a( 3 + 1) 2 We need to find PN, which is h 3 : PN = a( 3 + 1) 2 3 = ( 3 + 3 2 )a
Related: Mathematics — Heights and Distances · All PYQ Banks