An observer 1.5 m tall is 28.5 m away from a chimney. The angle of elevation of the top of the chimney from her eyes is 45°. What is the height of the chimney?
Step-by-Step Solution
Key Concept: Use the definition of tangent in a right‑angled triangle: \(\tan \theta = \dfrac{\text{opposite side}}{\text{adjacent side}}\). Here the opposite side is the vertical height of the chimney above the observer's eye level, and the adjacent side is the horizontal distance between the observer and the chimney.
1. Draw a right‑angled triangle
- Let point \(O\) be the observer's eye (1.5 m above the ground).
- Let point \(C\) be the top of the chimney.
- Let point \(B\) be the foot of the chimney on the ground.
- \(OB\) is the horizontal distance = 28.5 m.
- \(OC\) is the line of sight making an angle of elevation \(45^{\circ}\) with the horizontal.
- \(BC\) is the vertical height of the chimney above the ground (what we need).
2. Apply the tangent definition
\[\tan 45^{\circ} = \frac{\text{height of chimney above observer's eye}}{\text{horizontal distance}}\]
Since \(\tan 45^{\circ}=1\),
\[1 = \frac{BC - 1.5}{28.5}\]
where \(BC\) is the total height of the chimney and \(1.5\) m is the observer's height.
3. Solve for the unknown height
\[BC - 1.5 = 28.5 \times 1 = 28.5\]
\[BC = 28.5 + 1.5 = 30 \text{ m}\]
4. State the answer
The height of the chimney is \(30\) metres.
Correct Answer: 30 m