1. **Problem statement:** Pierre is on a viewing deck 300 m above the ground. He looks down at point A with an angle of depression of 40° and then further down at point B with an additional 32° angle of depression (total 72° from horizontal). We need to find the distance between points A and B on the ground. 2. **Understanding the problem:** The angles of depression correspond to angles between the horizontal line from Pierre's eye level and the lines of sight to points A and B. Since the tower is vertical, the ground is horizontal, and the angles of depression are angles between the horizontal and the line of sight downward. 3. **Key formula:** For a right triangle with height $h$ and angle of depression $\theta$, the horizontal distance $d$ from the base of the tower to the point on the ground is given by: $$ d = h \tan(\theta) $$ 4. **Calculate distances:** - Distance to A: $d_A = 300 \tan(40^\circ)$ - Distance to B: $d_B = 300 \tan(72^\circ)$ 5. **Calculate values:** - $\tan(40^\circ) \approx 0.8391$ - $\tan(72^\circ) \approx 3.0777$ So, $$ d_A = 300 \times 0.8391 = 251.73 \text{ m} $$ $$ d_B = 300 \times 3.0777 = 923.31 \text{ m} $$ 6. **Find distance between A and B:** Points A and B lie on the same horizontal line, so the distance between them is: $$ \text{Distance} = d_B - d_A = 923.31 - 251.73 = 671.58 \text{ m} $$ **Final answer:** The distance from Alain to Bailee is approximately **671.58 meters**.