Question

A tree grows vertically on a hillside. The hill is at a 16° angle to the horizontal. The tree casts an 18-meter shadow up the hill when the angle of elevation of the sun measures

68°. How tall is the tree?

Answer 1

The height of the tree on the hillside is calculated using trigonometry by splitting the height into two parts, resulting in a total height of 47.79 meters.

Step-by-step explanation:

To find the height of the tree growing on a hillside, we need to use trigonometric functions to split the tree's height into two parts: one parallel and one perpendicular to the hill's slope. The angle of the hill to the horizontal is given as 16°, and the angle of elevation of the sun is 68°.

The first step is to calculate the height of the part below the line parallel to the horizontal:

Height of the part below the line parallel to the horizontal = 18 sin16° = 4.96 meters

Next, we calculate the horizontal distance from the tip of the shadow to the tree:

Horizontal distance of the tip of the shadow from the tree = 18 cos16° = 17.30 meters

Finally, we calculate the height of the part above the line parallel to the horizontal:

Height of the part above the line parallel to the horizontal = 17.3 tan68° = 42.83 meters

By adding both parts, we determine the total height of the tree:

Height of the tree = 4.96 + 42.83 = 47.79 meters

Answer 2

The height of the tree is the sum of the part below the line parallel to the horizontal and the part above the line parallel to the horizontal.

Height of the part below the line parallel to the horizontal = 18 sin16° = 4.96 meters
Horizontal distance of the tip of the of the shadow from the tree = 18 cos16° = 17.30 meters
Height of the part above the line parallel to the horizontal = 17.3 tan68° = 42.83 meters

Height of the tree = 4.96 + 42.83 = 47.79 meters