On the same side of a tower, two objects are located. Observed from the top of the tower, their angles of depression are $45^\circ$ and $60^\circ$. If the height of the tower is 150 m, the distance between the objects is

### Concept & Distance Between Two Objects When two objects are on the same side of a vertical structure, the distance between them is the difference of their individual horizontal distances from the base of the structure. $$ \text{Distance} = \frac{h}{\tan(\theta_{\text{smaller}})} - \frac{h}{\tan(\theta_{\text{larger}})} $$ ### Step-by-Step Solution 1. Given: Height of the tower $h = 150$ m. The angles of depression (and thus angles of elevation) are $45^\circ$ and $60^\circ$. 2. Find the distance of the first object ($d_1$) using the $45^\circ$ angle: $$ \tan(45^\circ) = \frac{150}{d_1} \Rightarrow 1 = \frac{150}{d_1} \Rightarrow d_1 = 150 \text{ m} $$ 3. Find the distance of the second object ($d_2$) using the $60^\circ$ angle: $$ \tan(60^\circ) = \frac{150}{d_2} \Rightarrow \sqrt{3} = \frac{150}{d_2} \Rightarrow d_2 = \frac{150}{\sqrt{3}} $$ 4. Rationalize $d_2$: $$ d_2 = \frac{150\sqrt{3}}{3} = 50\sqrt{3} \text{ m} $$ 5. Calculate the distance between the two objects: $$ \text{Distance} = d_1 - d_2 = 150 - 50\sqrt{3} $$ 6. Substitute $\sqrt{3} \approx 1.73$ (as commonly used for exact approximations in these options): $$ \text{Distance} = 150 - 50(1.73) = 150 - 86.5 = 63.5 \text{ m} $$ ### Exam Strategy & Shortcut Use the direct formula for distance between objects on the same side: $d = h(\cot \theta_1 - \cot \theta_2)$. $d = 150(\cot 45^\circ - \cot 60^\circ) = 150(1 - \frac{1}{\sqrt{3}}) = 150 - 50\sqrt{3} = 150 - 86.6 \approx 63.5$. ### Common Pitfall Using $\tan$ instead of $\cot$ or multiplying the distances instead of subtracting them. Also, confusing whether to add or subtract; if they are on the "same side," you subtract. ### Final Answer Therefore, the correct answer is **63.5 m**.