Question 1

A plane is observed to be approaching the airport. It is at a distance of 10 km from the point of observation and makes an angle of elevation of 67 5^ . What is the height of the plane above the groun

Accepted Answer

Let h be the height of the plane above the ground. The distance from the point of observation to the plane is d = 10 km. The angle of elevation is = 67.5^ . From the right-angled triangle formed, = h d . h = 10 67.5^ Using the half-angle formula, = 1 - 2 2 . 67.5^ = 1 - 135^ 2 = 1 - (- 1 2 ) 2 = 2 + 1 2 2 Multiplying the numerator and denominator inside the square root by 2 : 67.5^ = 2 + 2 4 = 2 + 2 2 Substituting this value into the equation for h: h = 10 2 + 2 2 = 5 2 + 2 km

Question 2

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)

Accepted Answer

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

Question 3

A man at M, standing 100 m away from the base (P) of a chimney of height 50 m, observes the angle of elevation of the highest point (Q) of the smoke to be 45°. The highest point of the chimney is at R

Accepted Answer

Let PM be the distance of the man from the base of the chimney, so PM = 100 m. Let PR be the height of the chimney, so PR = 50 m. Let Q be the highest point of the smoke. The angle of elevation of Q from M is PMQ = 45^ . In PMQ, ( PMQ) = PQ PM (45^ ) = PQ 100 PQ = 100 m. In PMR, ( PMR) = PR PM = 50 100 = 1 2 . The required angle is RMQ = PMQ - PMR = 45^ - PMR. Taking tangent on both sides: ( RMQ) = (45^ - PMR) = (45^ ) - ( PMR) 1 + (45^ ) ( PMR) ( RMQ) = 1 - 1 2 1 + 1 1 2 = 1 2 3 2 = 1 3 RMQ = ^

Question 4

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)

Accepted Answer

Let MN = h be the height of the tower. The distances of points P, Q, and R from the foot of the tower N are given by: PN = h 30^ = h 3 QN = h 45^ = h RN = h 60^ = h 3 Given that P, Q, R lie on a straight line passing through N, the distance QR is: QR = QN - RN b = h - h 3 b = h ( 3 -1 3 ) h = b 3 3 -1 Rationalizing the denominator: h = b 3 ( 3 +1) ( 3 -1)( 3 +1) h = b(3+ 3 ) 3-1 h = ( 3+ 3 2 )b Answer: ( 3+ 3 2 )b

Heights and Distances — Previous Year Questions & Solutions