Question

When the angle of elevation of sun becomes \( \theta \) from \( \phi \), then the shadow of a pillar situated in a horizontal plane increases by a metre. Find the height of the pillar.

Hint:

Using the cotangent ratio (\(\cot = \text{Adjacent}/\text{Opposite}\)) can sometimes lead to simpler algebraic manipulation in heights and distances problems, especially when the adjacent side is the unknown you are trying to relate.

Answer 1

Step 1: Understanding the Concept:
As the sun's angle of elevation changes, the length of an object's shadow also changes. This problem relates the change in shadow length to the change in the angle of elevation using trigonometry. The shadow is longer when the angle of elevation is smaller, so we assume \( \phi>\theta \).
Step 2: Key Formula or Approach:
We will model the situation with two right-angled triangles and use the cotangent ratio: \( \cot \theta = \frac{\text{Adjacent}}{\text{Opposite}} \).
Step 3: Detailed Explanation:
Let PQ be the pillar of height \(h\).
Let QA be the initial shadow when the angle of elevation is \( \phi \), so \( \angle PAQ = \phi \).
Let QB be the final shadow when the angle of elevation is \( \theta \), so \( \angle PBQ = \theta \).
The shadow increases by 'a' metres, so AB = 'a'.
Let the length of the initial shadow QA = \(x\). Then the length of the final shadow QB = \(x + a\).
In the right-angled triangle \(\triangle PQA\):
\[ \cot \phi = \frac{QA}{PQ} = \frac{x}{h} \] \[ x = h \cot \phi \quad \ldots(1) \] In the right-angled triangle \(\triangle PQB\):
\[ \cot \theta = \frac{QB}{PQ} = \frac{x+a}{h} \] \[ x + a = h \cot \theta \quad \ldots(2) \] Substitute the value of \(x\) from equation (1) into equation (2):
\[ h \cot \phi + a = h \cot \theta \] Rearrange to solve for \(h\):
\[ a = h \cot \theta - h \cot \phi \] \[ a = h (\cot \theta - \cot \phi) \] \[ h = \frac{a}{\cot \theta - \cot \phi} \] Step 4: Final Answer:
The height of the pillar is \( \frac{a}{\cot \theta - \cot \phi} \) metres.