Wisden was crossing a bridge constructed on a river. While standing on the bridge, he saw 2 boats Liberty and Columbus cross from the opposite side of the river. He could see liberty at an angle of depression of 60° and Columbus at 45°.Wisden’ s height is 168cm and the height of the bridge from the ground is 170 m from the ground. Two people riding the boats saw the opposite edge of the bridge at an angle of 45. Find the distance between the 2 boats and also the distance to cover by the two boats to reach exactly below the bridge where Wisden is standing.

To solve this problem, we need to visualize the scenario involving angles of depression and some basic trigonometry. Wisden is standing on a bridge and observing two boats, Liberty and Columbus, at different angles. We can break this down step by step to find the distances between the boats and the distance each boat needs to travel to reach directly below Wisden.

Understanding the Setup

First, let’s clarify the heights involved. Wisden is 168 cm tall, which is 1.68 m. The bridge is 170 m above the ground. Therefore, the total height from which Wisden is observing the boats is:

Total Height = Height of Bridge - Height of Wisden

Total Height = 170 m - 1.68 m = 168.32 m

Calculating Distances Using Angles of Depression

The angles of depression from Wisden to the boats are given as 60° for Liberty and 45° for Columbus. The angle of depression is measured from the horizontal line down to the object. Thus, we can use these angles to find the horizontal distances from Wisden to each boat using the tangent function.

Distance to Liberty

For Liberty, using the angle of depression of 60°:

tan(60°) = Opposite / Adjacent

Here, the opposite side is the height from Wisden to the water level (which is 168.32 m), and the adjacent side is the distance from Wisden to Liberty (let's call this distance d₁):

tan(60°) = 168.32 / d₁

Rearranging gives:

d₁ = 168.32 / tan(60°)

Since tan(60°) = √3 (approximately 1.732), we have:

d₁ = 168.32 / 1.732 ≈ 97.1 m

Distance to Columbus

Now, for Columbus, with an angle of depression of 45°:

tan(45°) = Opposite / Adjacent

Again, the opposite side is the same height (168.32 m), and the adjacent side is the distance from Wisden to Columbus (let's call this d₂):

tan(45°) = 168.32 / d₂

Since tan(45°) = 1, we find:

d₂ = 168.32 / 1 = 168.32 m

Finding the Distance Between the Two Boats

Now that we have the distances from Wisden to each boat, we can find the distance between Liberty and Columbus. Since they are on opposite sides of Wisden, the total distance between the two boats is:

Distance between boats = d₁ + d₂

Distance between boats = 97.1 m + 168.32 m ≈ 265.42 m

Distance to Reach Below the Bridge

Finally, we need to determine how far each boat must travel to reach directly below Wisden. Since we already calculated the distances to each boat, they simply need to travel those distances:

• Liberty needs to cover approximately 97.1 m.
• Columbus needs to cover approximately 168.32 m.

In summary, the distance between the two boats is approximately 265.42 m, and Liberty and Columbus need to travel about 97.1 m and 168.32 m, respectively, to reach directly below Wisden on the bridge.