The top of a 15 metre high tower makes an angle of elevation of $60^\circ$ with the bottom of an electric pole and angle of elevation of $30^\circ$ with the top of the pole. What is the height of the electric pole?

### Concept & Multi-level Angles of Elevation This problem involves two right-angled triangles sharing the same horizontal base distance. By setting up an equation for the shared horizontal distance, we can solve for the unknown height. $$ x = \frac{\text{Height}_1}{\tan(\theta_1)} = \frac{\text{Height}_2}{\tan(\theta_2)} $$ ### Step-by-Step Solution 1. Let the height of the tower be $H = 15$ m. 2. Let the height of the electric pole be $h$. 3. Let the horizontal distance between the tower and the pole be $x$. 4. From the bottom of the electric pole, the angle of elevation to the top of the tower is $60^\circ$: $$ \tan(60^\circ) = \frac{15}{x} \Rightarrow \sqrt{3} = \frac{15}{x} \Rightarrow x = \frac{15}{\sqrt{3}} = 5\sqrt{3} \text{ m} $$ 5. From the top of the electric pole, the horizontal line to the tower is at a height $h$. The remaining height of the tower above the pole is $(15 - h)$. 6. The angle of elevation to the top of the tower from the top of the pole is $30^\circ$: $$ \tan(30^\circ) = \frac{15 - h}{x} $$ 7. Substitute $x = 5\sqrt{3}$ and $\tan(30^\circ) = \frac{1}{\sqrt{3}}$: $$ \frac{1}{\sqrt{3}} = \frac{15 - h}{5\sqrt{3}} $$ 8. Multiply both sides by $5\sqrt{3}$: $$ 5 = 15 - h $$ 9. Solve for $h$: $$ h = 15 - 5 = 10 \text{ m} $$ ### Exam Strategy & Shortcut Notice the angles $60^\circ$ and $30^\circ$. The tangents are $\sqrt{3}$ and $\frac{1}{\sqrt{3}}$. The ratio of the vertical heights they cover over the same horizontal distance $x$ will be $3:1$. Total height is 3 parts ($15$m). The top segment is 1 part ($5$m). The pole height is the remaining 2 parts ($10$m). ### Common Pitfall Misinterpreting the phrasing "makes an angle... with the bottom". It means the observer is at the bottom (or top) of the pole looking towards the top of the tower. Drawing the diagram backwards will yield an impossible answer. ### Final Answer Therefore, the correct answer is **10 metres**.