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. Further P, R and Q are in a straight line and the straight line is perpendicular to PM. What is the angle RMQ equal to?

Given:

• The height of the chimney (point Q) is given as 50 meters.
• The man is 100 meters away from the base of the chimney (point P).
• The angle of elevation from the man (at point M) to the top of the chimney (point Q) is 45^∘
• There is a straight line from point P through point R, and it is perpendicular to line PM.

Calculation:

For the right-angled triangle △ MPQ the tangent of the angle of elevation at point M is

⇒ tan 45^∘ = PQ/PM = \(\frac{50 + h }{100}\) 

where h is the additional height at point Q due to the smoke

Since tan 45∘ = 1

⇒ 1 = \(\frac{50 + h }{100} = 100 = 50 + h \)

⇒ h = 50 m

So, the height at point Q due to the smoke is 50 meters.

Finding the angle ∠RMQ

∠RMQ, is formed by the line PR (the straight line from P to R) and the line PM.

Using the definition of the tangent function, we have

⇒ tan ∠RMQ = \(\frac{50}{100} = \frac{1}{2}\)

⇒ ∠RMQ = tan^-1\(\frac{1}{2}\)

Hence, the Correct answer is Option 1.