The angle of depression of a point situated at a distance of 70m from the base of a tower is $60^\circ$. The height of the tower is

### Concept & Angle of Depression Equality Due to alternate interior angles, the angle of depression from the top of a tower to a point on the ground is exactly equal to the angle of elevation from that point on the ground to the top of the tower. $$ \text{Angle of Depression} = \text{Angle of Elevation} $$ ### Step-by-Step Solution 1. Given: Distance from the base ($x$) = $70$ m. Angle of depression = $60^\circ$. 2. The angle of elevation from the point to the top of the tower is therefore also $60^\circ$. 3. Let the height of the tower be $h$. 4. Apply the tangent function for the right-angled triangle: $$ \tan(60^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{h}{70} $$ 5. Substitute the known value $\tan(60^\circ) = \sqrt{3}$: $$ \sqrt{3} = \frac{h}{70} $$ 6. Multiply both sides by $70$ to solve for $h$: $$ h = 70\sqrt{3} \text{ m} $$ ### Exam Strategy & Shortcut In a $30^\circ$-$60^\circ$-$90^\circ$ triangle, the side opposite the $60^\circ$ angle is always $\sqrt{3}$ times the adjacent base. Given the base is $70$, the height is instantly $70\sqrt{3}$. ### Common Pitfall Incorrectly using the angle of depression as the angle at the top vertex inside the vertical triangle (which would actually be $90^\circ - 60^\circ = 30^\circ$), leading to the incorrect answer of $70/\sqrt{3}$ or $\frac{70\sqrt{3}}{3}$ m. ### Final Answer Therefore, the correct answer is **$70\sqrt{3}$ m**.