1. **Problem 7:** A flagpole creates a 7.9 m long shadow when the sun is at an angle of elevation of 59°. Find the height of the flagpole. 2. We use the tangent function from SOH CAH TOA: $$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\text{height}}{\text{shadow length}}$$ 3. Substitute the known values: $$\tan(59^\circ) = \frac{h}{7.9}$$ 4. Solve for height $h$: $$h = 7.9 \times \tan(59^\circ)$$ 5. Calculate: $$h = 7.9 \times 1.6643 = 13.15$$ 6. Rounded to one decimal place: $$h \approx 13.2\text{ m}$$ --- 7. **Problem 8a:** A 10 m ladder leans against a wall, with the base 6 m from the wall. Find the angle $D$ the ladder makes with the ground. 8. Use cosine from SOH CAH TOA: $$\cos(D) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{6}{10}$$ 9. Calculate the angle: $$D = \cos^{-1}\left(\frac{6}{10}\right) = \cos^{-1}(0.6)$$ 10. Evaluate: $$D \approx 53.13^\circ$$ --- 11. **Problem 8b:** Find how high up the wall the ladder reaches. 12. Use Pythagoras theorem or sine function. Using sine: $$\sin(D) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{10}$$ 13. Solve for $h$: $$h = 10 \times \sin(53.13^\circ)$$ 14. Calculate: $$h = 10 \times 0.8 = 8.0\text{ m}$$ **Final answers:** - Height of flagpole: $13.2$ m - Angle ladder makes with ground: $53.13^\circ$ - Height ladder reaches on wall: $8.0$ m