1. **Problem Statement:** From a window 16 ft above the ground, a student measures the angles of elevation and depression to the top and base of a nearby tree. We need to find: a) The horizontal distance between the student and the tree. b) The height of the tree. 2. **Given Data:** - Height of window above ground: $16$ ft - Angle of depression to base of tree: $40^\circ$ - Angle of elevation to top of tree: $16^\circ$ 3. **Formulas and Rules:** - Use right triangle trigonometry: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ - The horizontal distance from the student to the tree is the same for both angles. 4. **Step a) Find horizontal distance $x$:** - For the angle of depression $40^\circ$, the vertical leg is $16$ ft (height of window). - Using $\tan(40^\circ) = \frac{16}{x}$ - Solve for $x$: $$x = \frac{16}{\tan(40^\circ)}$$ - Calculate $\tan(40^\circ) \approx 0.8391$ - So, $$x = \frac{16}{0.8391} \approx 19.07 \text{ ft}$$ 5. **Step b) Find height of tree $h$:** - The total height of the tree is the height from ground to window plus the height from window to top of tree. - Let $h_1 = 16$ ft (window height), and $h_2$ be the height from window to top of tree. - Using angle of elevation $16^\circ$: $$\tan(16^\circ) = \frac{h_2}{x}$$ - Solve for $h_2$: $$h_2 = x \times \tan(16^\circ)$$ - Calculate $\tan(16^\circ) \approx 0.2867$ - So, $$h_2 = 19.07 \times 0.2867 \approx 5.47 \text{ ft}$$ - Total height: $$h = h_1 + h_2 = 16 + 5.47 = 21.47 \text{ ft}$$ 6. **Final answers rounded to nearest foot:** - Horizontal distance: $\boxed{19}$ ft - Height of tree: $\boxed{21}$ ft