Question

a drone flying at a constant speed of 11 m/s is currently located directly above a tree. the drone begins to descend towards a landing pad, as shown below. (note that the figure is not drawn to scale.) the tree and the landing pad are 420 m apart, and the drone’s angle of descent is 11° from the horizontal. find the remaining distance, r, that will be between the drone and the landing pad after 24 seconds. carry your intermediate computations to at least four decimal places, and round your answer to the nearest tenth of a meter.

Explanation:

Step1: Calculate distance drone travels

The drone's speed is \( 11 \, \text{m/s} \) and time is \( 24 \, \text{s} \). Using \( \text{distance} = \text{speed} \times \text{time} \), we get \( 11 \times 24 = 264 \, \text{m} \).

Step2: Find hypotenuse of big triangle

The horizontal distance between tree and landing pad is \( 420 \, \text{m} \). The big right triangle has angle \( 11^\circ \) and adjacent side \( 420 \, \text{m} \). Using \( \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} \), hypotenuse \( H = \frac{420}{\cos(11^\circ)} \). Calculate \( \cos(11^\circ) \approx 0.9816 \), so \( H \approx \frac{420}{0.9816} \approx 427.87 \, \text{m} \).

Step3: Subtract drone's travel from hypotenuse

The remaining distance \( r = H - 264 \). So \( r \approx 427.87 - 264 = 163.87 \)? Wait, no, wait. Wait, the big triangle: the vertical side is the height, horizontal is 420. Wait, maybe I mixed up. Wait, the drone is descending at \( 11^\circ \) from horizontal, so the path of the drone is the hypotenuse of a right triangle with horizontal leg 420 m (distance between tree and landing pad, which is horizontal). Wait, no, the tree is directly below the drone initially, so the horizontal distance from drone to landing pad is 420 m. So the drone's path to landing pad is a hypotenuse \( L \) where \( \cos(11^\circ) = \frac{420}{L} \), so \( L = \frac{420}{\cos(11^\circ)} \approx \frac{420}{0.9816271834} \approx 427.868 \, \text{m} \). Then the drone travels \( 11 \times 24 = 264 \, \text{m} \) along this path. So remaining distance \( r = L - 264 \approx 427.868 - 264 = 163.868 \), rounded to nearest tenth is \( 163.9 \)? Wait, no, maybe I made a mistake. Wait, let's re-examine.

Wait, the problem says "the tree and the landing pad are 420 m apart" (horizontal distance, since drone is above tree, so horizontal distance between tree and landing pad is 420 m). The drone's angle of descent is \( 11^\circ \) from the horizontal, so the path of the drone to the landing pad is a straight line (hypotenuse) where the horizontal component is 420 m. So the length of the path from drone's initial position to landing pad is \( \frac{420}{\cos(11^\circ)} \). Then, the drone flies for 24 seconds at 11 m/s, so distance flown is \( 11 \times 24 = 264 \) m. Therefore, remaining distance \( r = \frac{420}{\cos(11^\circ)} - 264 \).

Calculating \( \cos(11^\circ) \approx 0.981627 \), so \( \frac{420}{0.981627} \approx 427.868 \). Then \( 427.868 - 264 = 163.868 \), which rounds to 163.9 meters. Wait, but let's check with sine. Wait, maybe the vertical side? No, angle is from horizontal, so adjacent is horizontal (420 m), hypotenuse is path length. So yes, that's correct.

Wait, but let's do the calculation more accurately.

First, distance traveled by drone: \( 11 \times 24 = 264 \) m.

Length of path from drone to landing pad: \( \frac{420}{\cos(11^\circ)} \).

Compute \( \cos(11^\circ) \):

\( \cos(11^\circ) \approx \cos(11) \approx 0.9816271834476643 \)

So \( \frac{420}{0.9816271834476643} \approx 420 \div 0.9816271834476643 \approx 427.868067 \)

Subtract distance traveled: \( 427.868067 - 264 = 163.868067 \)

Round to nearest tenth: \( 163.9 \) meters.

Wait, but maybe I messed up the triangle. Let's think again. The drone is above the tree, landing pad is 420 m from tree (horizontal). So the horizontal distance between drone (initial) and landing pad is 420 m. The drone descends at \( 11^\circ \) from horizontal, so the path to landing pad is a straight line with horizontal component 420 m. So the length of that path is \( 420 / \cos(11^\circ) \). Then, after flying 264 m along this path, the remaining distance is that length minus 264. So that's correct.

Answer:

\( 163.9 \) (rounded to nearest tenth)