Question

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 swing set is 31.43°, and emir is 11 feet from the ground. how many feet is the base of the tree from the swingset? round your answer to the nearest foot.
15 feet
20 feet
18 feet
24 feet

Explanation:

Step1: Identify the trigonometric relationship

We have a right triangle where the opposite side (height from Emir's eyes to ground, but wait, Emir is 11 feet from the ground, and the angle of depression is 31.43°. The angle of depression is equal to the angle of elevation from the swing set to Emir's eyes. So we can use the tangent function: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$. Here, the opposite side is 11 feet (height of Emir from ground), and the adjacent side is the distance we need to find (let's call it $x$), and $\theta = 31.43^\circ$. Wait, no, wait: angle of depression is from Emir's eyeline to the swing set. So the triangle has the vertical side as 11 feet (Emir's height from ground), the horizontal side as $x$ (distance from tree base to swing set), and the angle at Emir's eyes (angle of depression) is equal to the angle at the swing set (angle of elevation) because they are alternate interior angles. So $\tan(31.43^\circ) = \frac{11}{x}$? Wait, no, wait: if Emir is looking down, the vertical side is 11 feet, and the horizontal side is $x$. Wait, maybe I got it reversed. Let's think again. The angle of depression is 31.43°, so the angle between Emir's horizontal eyeline and the line of sight to the swing set is 31.43°. So the triangle formed has: vertical leg = 11 feet (from ground to Emir's eyes), horizontal leg = $x$ (distance from tree base to swing set), and the angle at Emir's eyes (between horizontal and line of sight) is 31.43°. So $\tan(31.43^\circ) = \frac{11}{x}$? Wait, no, $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$. If the angle is at Emir's eyes, then the opposite side is the vertical distance (11 feet), and the adjacent side is the horizontal distance ($x$). So $\tan(31.43^\circ) = \frac{11}{x}$, so $x = \frac{11}{\tan(31.43^\circ)}$. Wait, but let's calculate $\tan(31.43^\circ)$. Let's check: $\tan(31.43^\circ) \approx \tan(31.43) \approx 0.6$. Wait, 11 divided by 0.6 is about 18.33, which is close to 18. Wait, maybe I made a mistake. Wait, maybe the angle of depression is such that the vertical side is the height, and the horizontal side is the distance. Wait, let's re-express:

Angle of depression = 31.43°, so the angle between the horizontal (Emir's eyeline) and the line of sight (to swing set) is 31.43°. So the triangle has:

- Vertical side: 11 feet (Emir's height from ground)
- Horizontal side: $x$ (distance from tree base to swing set)
- Angle at Emir's eyes: 31.43°

So $\tan(31.43^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{11}{x}$? Wait, no, opposite would be the vertical side, adjacent is the horizontal side. So $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{11}{x}$, so $x = \frac{11}{\tan(31.43^\circ)}$. Let's calculate $\tan(31.43^\circ)$. Let's use a calculator: 31.43 degrees. $\tan(31.43) \approx \tan(31.43) \approx 0.600$ (since $\tan(30^\circ) \approx 0.577$, $\tan(35^\circ) \approx 0.700$, so 31.43 is close to 31.4, which is $\pi/6$? No, 31.43 degrees. Let's calculate: 31.43 * $\pi$/180 ≈ 0.548 radians. $\tan(0.548) ≈ 0.600$. So 11 / 0.6 ≈ 18.33, which rounds to 18. So the distance is approximately 18 feet.

Step2: Calculate the distance

We use the tangent function: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$. Here, $\theta = 31.43^\circ$, opposite = 11 feet, adjacent = $x$ (distance we need to find). So:

$\tan(31.43^\circ) = \frac{11}{x}$

Solving for $x$:

$x = \frac{11}{\tan(31.43^\circ)}$

Using a calculator to find $\tan(31.43^\circ) \approx 0.6$ (more accurately, let's calculate $\tan(31.43^\circ)$:

31.43 degrees. Let's use a calculator: $\tan(31.43) \approx 0.600$ (exact value: let's check with calculator input: 31.43, tan. Let's assume it's approximately 0.600). Then:

$x = \frac{11}{0.6} \approx 18.33$, which rounds to 18.

Answer:

18 feet