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 $45^\circ$. What is the height of the chimney?

Step 1: Define the geometry

We first define the elements of the problem as described in the question. Let AB represent the height of the chimney, and CD represent the height of the observer. The observer is standing on the ground, and the chimney is perpendicular to the ground.

Step 2: Identify given values

The problem provides us with the height of the observer as $1.5 \text{ m}$. The horizontal distance between the observer and the chimney is given as $28.5 \text{ m}$. The angle of elevation from the observer's eyes to the top of the chimney is $45^\circ$.

Step 3: Form a right-angled triangle

To use trigonometry, we construct a right-angled triangle. We draw a line DE from the observer's eye (D) parallel to the ground CB'. This line forms a right angle at point E, where E is on the chimney AB. So, DE will be equal to the distance from the chimney, $28.5 \text{ m}$, and EB will be equal to the height of the observer, $1.5 \text{ m}$.

Step 4: Use trigonometric ratio in $\triangle$ ADE

In the right-angled triangle ADE, the angle of elevation is $\angle\text{ADE} = 45^\circ$. We need to find the height AE. The tangent function relates the opposite side (AE) to the adjacent side (DE) with respect to the angle $\angle\text{ADE}$.

Step 5: Substitute values and solve for AE

Substitute the known values into the tangent equation. We know that $\tan(45^\circ) = 1$. Solving for AE, we find that the height from the observer's eyes to the top of the chimney is $28.5 \text{ m}$.

Step 6: Calculate the total height of the chimney

The total height of the chimney AB is the sum of the height AE, which we just calculated, and the height EB, which is the height of the observer. Adding these two values gives us the total height of the chimney.

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