1. **Problem 4: Find the angle of elevation from the device to the top of the tree.** Given: - Distance from surveyor to tree (adjacent side) = 100 feet - Height of device above ground = 5 feet - Total height of tree = 133 feet We want the angle of elevation $\theta$ from the device to the top of the tree. 2. **Calculate the effective height difference (opposite side):** $$\text{opposite} = 133 - 5 = 128 \text{ feet}$$ 3. **Use the tangent function for right triangles:** $$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{128}{100}$$ 4. **Calculate the angle $\theta$ by taking the arctangent:** $$\theta = \tan^{-1}\left(\frac{128}{100}\right)$$ 5. **Evaluate the arctangent:** $$\theta \approx \tan^{-1}(1.28) \approx 52.43^\circ$$ --- 1. **Problem 5: Find the angle of elevation of the ladder.** Given: - Ladder length (hypotenuse) = 12 feet - Height up the wall (opposite side) = 10.1 feet We want the angle of elevation $\alpha$ between the base and the ladder. 2. **Use the sine function:** $$\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{10.1}{12}$$ 3. **Calculate the angle $\alpha$ by taking the arcsine:** $$\alpha = \sin^{-1}\left(\frac{10.1}{12}\right)$$ 4. **Evaluate the arcsine and round to 3 decimal places:** $$\alpha \approx \sin^{-1}(0.8417) \approx 57.096^\circ$$ --- 1. **Problem 6: Find the maximum height of the ramp so the angle does not exceed 10 degrees.** Given: - Base (adjacent side) = 20 feet - Maximum angle $\beta = 10^\circ$ We want the maximum height (opposite side) $h$. 2. **Use the tangent function:** $$\tan(\beta) = \frac{h}{20}$$ 3. **Solve for $h$:** $$h = 20 \times \tan(10^\circ)$$ 4. **Evaluate and round to 3 decimal places:** $$h \approx 20 \times 0.1763 = 3.526$$ --- **Final answers:** - Problem 4 angle of elevation: $52.43^\circ$ - Problem 5 angle of elevation: $57.096^\circ$ - Problem 6 maximum ramp height: $3.526$ feet