Emir is standing in a treehouse and looking down at a swingset in the yard next door. The angle of depression from Emir's eyeline to the swingset is 30.26°, and Emir is 14 feet from the ground. How many feet is the base of the tree from the swingset? Round your answer to the nearest foot. (4 points)

Question

15 feet

20 feet

18 feet

24 feet

Answer 1

To find the distance from the base of the tree to the swingset, we can use the angle of depression and the height of the treehouse.

1. Identify the triangle: When Emir looks down at the swingset, a right triangle is formed where:

   • The height of the treehouse (vertical leg) = 14 feet (the height at which Emir is standing).
   • The angle of depression to the swingset = 30.26°.
   • The horizontal leg (the distance from the base of the tree to the swingset) is what we need to find. Let's denote this distance as \( d \).

2. Use trigonometry: We use the tangent function, which relates the angle (θ) to the opposite side (height) and the adjacent side (distance from the tree). The formula is: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \] Plugging in our values, we have: \[ \tan(30.26°) = \frac{14}{d} \]

3. Rearranging the equation: \[ d = \frac{14}{\tan(30.26°)} \]

4. Calculating tan(30.26°): Using a calculator, we find: \[ \tan(30.26°) \approx 0.582 \]

5. Calculating d: \[ d \approx \frac{14}{0.582} \approx 24.05 \text{ feet} \]

6. Rounding to the nearest foot: \[ d \approx 24 \text{ feet} \]

Thus, the base of the tree is approximately 24 feet from the swingset. The correct answer is:

24 feet.