Question

From top of view point at height of \(80\,\text{m}\), the angles of depression of the top and bottom of a flag standing on the same plane are observed to be \(30^\circ\) and \(45^\circ\). Find the height of the flag.

• A. \(40\sqrt3\)
• B. \(\frac{80}{\sqrt3}\)
• C. \(80\left(1-\frac1{\sqrt3}\right)\)
• D. \(80(\sqrt3-1)\)

Hint:

For angle of depression problems, first draw horizontal lines and remember that angle of depression equals angle of elevation. Then apply the tangent ratio: \[ \tan\theta=\frac{\text{opposite}}{\text{adjacent}}. \]

Answer 1

The Correct Option is C

Concept: The angle of depression from an elevated point is equal to the corresponding angle of elevation from the object. Using right-angled triangles and tangent ratios, we can determine the horizontal distance and then the height of the flag.

Step 1: Find the horizontal distance from the observation point to the flag.
Let the horizontal distance be \(d\). The angle of depression to the bottom of the flag is \[ 45^\circ. \] Hence, \[ \tan45^\circ = \frac{80}{d}. \] Since \[ \tan45^\circ=1, \] we get \[ d=80. \]

Step 2: Let the height of the flag be \(h\).
The angle of depression to the top of the flag is \[ 30^\circ. \] Therefore, \[ \tan30^\circ = \frac{80-h}{80}. \] Using \[ \tan30^\circ=\frac1{\sqrt3}, \] we obtain \[ \frac1{\sqrt3} = \frac{80-h}{80}. \]

Step 3: Solve for \(h\).
Multiplying by \(80\), \[ 80-h = \frac{80}{\sqrt3}. \] Therefore, \[ h = 80-\frac{80}{\sqrt3}. \] \[ h = 80\left(1-\frac1{\sqrt3}\right). \] Hence the height of the flag is \[ \boxed{80\left(1-\frac1{\sqrt3}\right)}. \]

Step 4: Interpretation of the result.
Since the angle to the top of the flag is smaller than the angle to the bottom, the top is closer to the observer's horizontal line. Therefore the flag must have a positive height less than \(80\) m, which is consistent with the obtained result.