Question

From the top of a cliff 50 m high, the angles of depression of the top and bottom of a tower are observed to be \(30^\circ\) and \(45^\circ\). The height of the tower is:

• A. \( 50\sqrt{3} \) m
• B. \( 50(\sqrt{3} - 1) \) m
• C. \( 50\left( 1 - \frac{\sqrt{3}}{3} \right) \) m
• D. 50(1 - √3/3) m

Hint:

Trigonometry is useful in solving real-world height and distance problems.

Answer 1

The Correct Option is D

Let the height of the tower be \( h \). Using the tangent function in \( \triangle ABD \): \[ \tan 45^\circ = \frac{AB}{BD} \Rightarrow BD = 50 { m} \] Now, in \( \triangle ACC' \): \[ \tan 30^\circ = \frac{AC'}{C'C} \] \[ \frac{1}{\sqrt{3}} = \frac{50 - h}{50} \] Solving for \( h \): \[ 50 = 50\sqrt{3} - h\sqrt{3} \] \[ h\sqrt{3} = 50(\sqrt{3} - 1) \] \[ h = 50 \left( 1 - \frac{\sqrt{3}}{3} \right) { m} \]

Answer 2

Step 1: Understanding the problem
We are given a cliff that is 50 meters high, and we observe the angles of depression from the top of the cliff to the top and bottom of a tower. The angles of depression are \( 30^\circ \) for the top of the tower and \( 45^\circ \) for the bottom of the tower. We are asked to find the height of the tower.

Step 2: Set up the situation geometrically
Let the height of the tower be \( h \), and let the horizontal distance from the cliff to the base of the tower be \( x \). We will assume the bottom of the tower is on the same level as the base of the cliff.
- The top of the tower has an angle of depression of \( 30^\circ \). Therefore, the line of sight from the top of the cliff to the top of the tower forms a right triangle. - The bottom of the tower has an angle of depression of \( 45^\circ \), so this also forms another right triangle. Let’s use trigonometric ratios to express the relationships.

Step 3: Use the angle of depression to set up equations
For the top of the tower, the angle of depression is \( 30^\circ \). Using the tangent function, we have: \[ \tan 30^\circ = \frac{50}{x} \] We know that \( \tan 30^\circ = \frac{1}{\sqrt{3}} \), so: \[ \frac{1}{\sqrt{3}} = \frac{50}{x} \quad \Rightarrow \quad x = 50\sqrt{3} \] For the bottom of the tower, the angle of depression is \( 45^\circ \). Using the tangent function again: \[ \tan 45^\circ = \frac{50}{x} \] Since \( \tan 45^\circ = 1 \), we have: \[ 1 = \frac{50}{x} \quad \Rightarrow \quad x = 50 \] Step 4: Height of the tower
Now that we know the horizontal distance \( x \), we can calculate the height of the tower using the two relationships. The height \( h \) of the tower is the difference between the heights corresponding to the two angles of depression.
Let \( h_1 \) be the height at the top of the tower and \( h_2 \) be the height at the bottom of the tower. From the previous results: \[ h_1 = 50 - 50\sqrt{3}/3 \quad \Rightarrow \quad \text{Height of the tower} = 50\left( 1 - \frac{\sqrt{3}}{3} \right) \] Step 5: Final Answer
The height of the tower is:

50(1 - \( \frac{\sqrt{3}}{3} \)) m