1. **State the problem:** Emma is 10 feet from the base of a tree and estimates the angle of elevation to the top is between 55° and 75°. We want to check if given height estimates are reasonable. 2. **Formula used:** The height $h$ of the tree can be found using the tangent function in a right triangle: $$h = d \times \tan(\theta)$$ where $d=10$ feet is the distance from the tree and $\theta$ is the angle of elevation. 3. **Calculate minimum and maximum height:** - For $\theta = 55^\circ$: $$h_{min} = 10 \times \tan(55^\circ)$$ - For $\theta = 75^\circ$: $$h_{max} = 10 \times \tan(75^\circ)$$ 4. **Evaluate tangent values:** - $\tan(55^\circ) \approx 1.428$ - $\tan(75^\circ) \approx 3.732$ 5. **Calculate heights:** $$h_{min} = 10 \times 1.428 = 14.28$$ $$h_{max} = 10 \times 3.732 = 37.32$$ 6. **Interpretation:** The tree height should be between 14.28 feet and 37.32 feet. 7. **Check each estimate:** - 4.2 feet: Not reasonable (less than 14.28) - 14.7 feet: Reasonable (between 14.28 and 37.32) - 24.4 feet: Reasonable - 33.9 feet: Reasonable - 39.1 feet: Not reasonable (greater than 37.32) - 58.7 feet: Not reasonable **Final answer:** Reasonable estimates are 14.7, 24.4, and 33.9 feet. Others are not reasonable.