Question

The shadow of a tower standing on a level plane is \(30\) m longer when sun’s elevation changes from \(60^\circ\) to \(30^\circ\). Find the height of the tower.

• A. \(10\sqrt{3}\) meter
• B. \(15\) meter
• C. \(30\sqrt{3}\) meter
• D. \(15\sqrt{3}\) meter

Hint:

Remember:

• \[ \tan\theta = \frac{\text{Height}}{\text{Shadow}} \]
• \[ \tan60^\circ = \sqrt{3} \]
• \[ \tan30^\circ = \frac{1}{\sqrt{3}} \]

Answer 1

The Correct Option is D

Concept: For a tower of height \(h\) and shadow length \(x\): \[ \tan\theta = \frac{h}{x} \] Thus: \[ x = \frac{h}{\tan\theta} \] As the angle of elevation decreases, the shadow becomes longer.

Step 1: Find the shadow length when angle is \(60^\circ\). Let height of tower be: \[ h \] Let shadow length at \(60^\circ\) be: \[ x_1 \] Using: \[ \tan60^\circ = \frac{h}{x_1} \] \[ \sqrt{3} = \frac{h}{x_1} \] \[ x_1 = \frac{h}{\sqrt{3}} \]

Step 2: Find the shadow length when angle is \(30^\circ\). Let shadow length at \(30^\circ\) be: \[ x_2 \] Using: \[ \tan30^\circ = \frac{h}{x_2} \] \[ \frac{1}{\sqrt{3}} = \frac{h}{x_2} \] \[ x_2 = h\sqrt{3} \]

Step 3: Use the condition on difference of shadows. Given: \[ x_2 - x_1 = 30 \] Substituting values: \[ h\sqrt{3} - \frac{h}{\sqrt{3}} = 30 \] Taking LCM: \[ \frac{3h-h}{\sqrt{3}} = 30 \] \[ \frac{2h}{\sqrt{3}} = 30 \] \[ 2h = 30\sqrt{3} \] \[ h = 15\sqrt{3} \] Therefore, \[ \boxed{15\sqrt{3}\text{ meter}} \]