1. **State the problem:** Two poles stand opposite each other across a road 80 m wide. One pole's height is \( \frac{3}{4} \) of the other's. From a certain point between them on the road, the angles of elevation to the tops of the poles are 60° and 30°. Find the heights of both poles and the distances from that point to each pole. 2. **Assign variables:** Let the height of the taller pole be \( H \). Then the shorter pole's height is \( \frac{3}{4}H \). Let the distance from the point on the road to the taller pole be \( x \). Then the distance to the shorter pole is \( 80 - x \). 3. **Use tangent of angles of elevation:** From the point, the angle of elevation to the taller pole's top is 60°, so $$ \tan 60^\circ = \frac{\text{height of taller pole}}{\text{distance to taller pole}} = \frac{H}{x} $$ And the angle to the shorter pole is 30°, so $$ \tan 30^\circ = \frac{\frac{3}{4}H}{80 - x} $$ 4. **Recall tangent values:** $$ \tan 60^\circ = \sqrt{3} $$ $$ \tan 30^\circ = \frac{1}{\sqrt{3}} $$ 5. **Write equations:** $$ \sqrt{3} = \frac{H}{x} \implies H = x \sqrt{3} $$ $$ \frac{1}{\sqrt{3}} = \frac{\frac{3}{4}H}{80 - x} \implies \frac{1}{\sqrt{3}} = \frac{3H}{4(80 - x)} $$ 6. **Substitute \( H = x \sqrt{3} \) into second equation:** $$ \frac{1}{\sqrt{3}} = \frac{3 (x \sqrt{3})}{4(80 - x)} = \frac{3x\sqrt{3}}{4(80 - x)} $$ 7. **Multiply both sides by denominator:** $$ (80 - x) \times \frac{1}{\sqrt{3}} = \frac{3x\sqrt{3}}{4} $$ 8. **Multiply both sides by \( 4 \sqrt{3} \):** $$ 4 \sqrt{3} (80 - x) \times \frac{1}{\sqrt{3}} = 4 \sqrt{3} \times \frac{3x \sqrt{3}}{4} $$ Simplifies to $$ 4 (80 - x) = 3x \times 3 $$ $$ 320 - 4x = 9x $$ 9. **Solve for \( x \):** $$ 320 = 9x + 4x = 13x $$ $$ x = \frac{320}{13} \approx 24.62 \, \text{m} $$ 10. **Find the height \( H \):** $$ H = x \sqrt{3} = \frac{320}{13} \times 1.732 \approx 42.68 \, \text{m} $$ 11. **Find shorter pole's height:** $$ \frac{3}{4} H = \frac{3}{4} \times 42.68 \approx 32.01 \, \text{m} $$ 12. **Find distance from point to shorter pole:** $$ 80 - x = 80 - 24.62 = 55.38 \, \text{m} $$ **Final answers:** - Taller pole height: approximately \( 42.68 \) m - Shorter pole height: approximately \( 32.01 \) m - Distance from point to taller pole: approximately \( 24.62 \) m - Distance from point to shorter pole: approximately \( 55.38 \) m