There is a palm tree on the bank of a river. Directly opposite it, on the other side of the river, a pole is fixed into the ground. If someone walks 7\(\sqrt{3}\) meters along the riverbank from the pole toward the tree, the base of the tree appears to form a 60° angle with the riverbank at that point. Determine the width of the river.

Q.There is a palm tree on the bank of a river. Directly opposite it, on the other side of the river, a pole is fixed into the ground. If someone walks 7\(\sqrt{3}\) meters along the riverbank from the pole toward the tree, the base of the tree appears to form a 60° angle with the riverbank at that point. Determine the width of the river.

Let’s assume point A is where the palm tree is located, and point B is where the pole is fixed on the opposite bank of the river. From point B, moving along the riverbank a distance of \(BC = 7\sqrt{3}\) meters, the base of the tree at point A forms an angle of ∠BCA = 60° with the riverbank. ∴ From triangle ABC, with respect to ∠BCA, we get: \(\tan∠BCA = \tan 60^\circ = \cfrac{AB}{BC} = \cfrac{AB}{7\sqrt{3}}\) Or, \(\tan 60^\circ = \cfrac{AB}{7\sqrt{3}}\) Or, \(\sqrt{3} = \cfrac{AB}{7\sqrt{3}}\) Or, \(AB = \sqrt{3} \times 7\sqrt{3}\) Or, \(AB = 21\) ∴ The river is 21 meters wide.