Midpoint Review Name: \( \qquad \) Shiloh.y Date: \( \qquad \) \( 03 / 19 / 25 \) Instructor: \( \qquad \) College Graduation Year: \( \qquad \) 2033 1. A bird is cruising at 80 meters per second at an altitude of 9,000 meters. If the plane starts to land with a \( 24^{\circ} \) amgle of depression, how long will it take to land? Round your answer to the nearest hundredth of a minute. *Non-negotiables*: - Draw and label applicable models with units - Find the distance of the bird's path to the ground - Use the distance you found (and the speed given) to find the time (use \( d=r t \) formula) - Use dimensional analysis to make sure you have the correct units - Round appropriately

To tackle this problem, let's break it down step by step while drawing the relevant model for better understanding. ### Step 1: Draw and Label the Model Imagine a right triangle where: - The height (altitude of the bird) is the vertical side, 9,000 meters. - The angle of depression from the bird to the ground is \( 24^\circ \). - The hypotenuse will represent the path of the bird to the ground. ### Step 2: Find the Distance of the Bird's Path to the Ground Using trigonometry, we can find the hypotenuse (distance along the bird's path) \( d \): \[ \sin(24^{\circ}) = \frac{\text{Opposite (Altitude)}}{\text{Hypotenuse (d)}} \] This means: \[ d = \frac{9000}{\sin(24^{\circ})} \approx \frac{9000}{0.4067} \approx 22,117.78 \text{ meters} \] ### Step 3: Calculate Time Using the Distance and Speed Now, using the distance we found and the speed of the bird, we can calculate the time taken to land: \[ d = r \cdot t \implies t = \frac{d}{r} \] Where: - \( d \) = 22,117.78 meters - \( r \) = 80 meters/second Substituting these values in gives: \[ t = \frac{22,117.78}{80} \approx 276.47 \text{ seconds} \] ### Step 4: Convert Seconds to Minutes To convert seconds to minutes, divide by 60: \[ t \approx \frac{276.47}{60} \approx 4.61 \text{ minutes} \] **Final Answer:** It will take approximately **4.61 minutes** for the bird to land, rounded to the nearest hundredth of a minute. ### Dimensional Analysis - Initial units of distance: meters - Speed: meters/second - Time calculated initially in seconds and then converted to minutes, confirming unit consistency throughout. By following these structured steps and keeping track of units, we ensure accuracy and clarity in our calculations!