Question

A dog chased a cat up a tree. The dog is 14 feet from the tree. The angle of depression from the cat to the dog is 36 degrees. How far is the cat up the tree? Round to the nearest tenth.

Answer 1

The cat is approximately 10.2 feet up the tree, which we calculated using the tangent function of trigonometry by relating the angle of depression and the adjacent side to the opposite side, which is the height of the cat.

Step-by-step explanation:

To find out how far the cat is up the tree, we can use trigonometry, specifically the tangent function which relates the angle of elevation or depression to the opposite and adjacent sides of a right triangle. The angle of depression from the cat to the dog being 36 degrees is equivalent to an angle of elevation of 36 degrees from the ground to the cat. The distance from the dog to the tree will be the adjacent side, and the height of the cat up the tree is the opposite side we need to find.

Using the tangent of the angle, we have:
tan(36 degrees) = opposite/adjacent
which translates to:
tan(36 degrees) = height of cat/14 feet.

Therefore, the height of the cat up the tree is:
14 feet * tan(36 degrees).

Using a calculator, we find that tan(36 degrees) is approximately 0.7265. Multiplying this by 14 gives us the cat's height of roughly 10.2 feet. We round it to the nearest tenth as per the question's instructions.