From the top of a tree of height 13 m the angle of elevation and depression of the top and bottom of another tree are 45° and 30° respectively. Find the height of the second tree.  (Give your answer to 2 decimal places)

To solve this problem, we will use the trigonometric ratios of the angles of elevation and depression.

Step 1: From the top of the first tree, the angle of elevation to the top of the second tree is 45°. This means that the height of the second tree above the first tree is equal to the distance between the two trees. Let's denote this distance as 'd'. Since the angle of elevation is 45°, the tangent of this angle is equal to the height divided by the base. In this case, the height and the base are the same (since the angle is 45°), so d = tan(45°) * d, which simplifies to d = d.

Step 2: From the top of the first tree, the angle of depression to the bottom of the second tree is 30°. This means that the height of the first tree above the ground is equal to the distance between the two trees divided by the tangent of the angle of depression. So, 13 = d / tan(30°). Solving for d, we get d = 13 * tan(30°) = 13 * √3/3 = 7.54 m (rounded to two decimal places).

Step 3: The height of the second tree is the height of the first tree plus the distance 'd'. So, the height of the second tree is 13 + 7.54 = 20.54 m.

So, the height of the second tree is 20.54 m.