From a ghat on one side of a 600-meter wide river, two boats set off toward the opposite bank in two different directions. If the first boat moves at an angle of 30° with the riverbank, and the second boat moves at a 90° angle to the path of the first boat, then what will be the distance between the two boats after they reach the opposite bank?

Q.From a ghat on one side of a 600-meter wide river, two boats set off toward the opposite bank in two different directions. If the first boat moves at an angle of 30° with the riverbank, and the second boat moves at a 90° angle to the path of the first boat, then what will be the distance between the two boats after they reach the opposite bank?

Let’s assume two boats set off from point A on the bank MN of a river that is 600 meters wide. The first boat moves at an angle of 30° with the riverbank and reaches point P on the opposite bank XY. The second boat moves at a 90° angle to the path of the first boat and reaches point Q on the same bank. A perpendicular AB is drawn from point A to the opposite bank XY, so AB = 600 meters. Given: ∠PAN = 30°, ∠PAQ = 90° ∴ ∠PAB = 60° and ∠QAB = 30° Now, from the right-angled triangle △PAB: \[ \frac{BP}{AB} = \tan 60^\circ = \sqrt{3} \Rightarrow BP = \sqrt{3} \cdot AB = 600\sqrt{3} \text{ meters} \] From the right-angled triangle △BAQ: \[ \frac{BQ}{AB} = \tan 30^\circ = \frac{1}{\sqrt{3}} \Rightarrow BQ = \frac{AB}{\sqrt{3}} = \frac{600}{\sqrt{3}} = \frac{600 \cdot \sqrt{3}}{3} = 200\sqrt{3} \text{ meters} \] Therefore, \[ PQ = BP + BQ = 600\sqrt{3} + 200\sqrt{3} = 800\sqrt{3} \text{ meters} \] So, the distance between the two boats after reaching the opposite bank is \(800\sqrt{3}\) meters.