Question

When the angle of elevation of the sun becomes \( \theta \) from \( \varphi \), then the shadow of a pillar standing on the horizontal ground is increased by \( \alpha \) metres. Find the length of the pillar.

Hint:

When dealing with problems involving angles of elevation and shadows, use the tangent function to relate the height of the object and the length of the shadow.

Answer 1

Let the height of the pillar be \( h \) metres and the initial length of the shadow be \( x_1 \) metres when the angle of elevation of the sun is \( \varphi \). We can use the tangent of the angle to relate the height and the length of the shadow: \[ \tan(\varphi) = \frac{h}{x_1}. \] Thus, the initial length of the shadow is: \[ x_1 = \frac{h}{\tan(\varphi)}. \] When the angle of elevation becomes \( \theta \), the length of the shadow increases by \( \alpha \) metres. Therefore, the new length of the shadow is \( x_2 = x_1 + \alpha \). Using the tangent function again for the new angle of elevation \( \theta \), we have: \[ \tan(\theta) = \frac{h}{x_2} = \frac{h}{x_1 + \alpha}. \] Step 1: Relate the two equations
From the first equation, \( x_1 = \frac{h}{\tan(\varphi)} \), substitute this into the second equation: \[ \tan(\theta) = \frac{h}{\frac{h}{\tan(\varphi)} + \alpha}. \] Step 2: Solve for \( h \)
Simplifying the equation: \[ \tan(\theta) = \frac{h}{\frac{h}{\tan(\varphi)} + \alpha}, \] \[ \tan(\theta) \left( \frac{h}{\tan(\varphi)} + \alpha \right) = h, \] \[ \tan(\theta) \cdot \frac{h}{\tan(\varphi)} + \tan(\theta) \cdot \alpha = h. \] Rearranging: \[ h \left( \tan(\theta) \cdot \frac{1}{\tan(\varphi)} - 1 \right) = - \tan(\theta) \cdot \alpha. \] Solving for \( h \): \[ h = \frac{- \tan(\theta) \cdot \alpha}{\left( \tan(\theta) \cdot \frac{1}{\tan(\varphi)} - 1 \right)}. \] This equation gives the length of the pillar in terms of the angles \( \theta \), \( \varphi \), and the increase in the shadow \( \alpha \).
Conclusion:
The length of the pillar is given by the formula derived above.