1. **Problem Statement:** A dragon at point D breathes fire at point H with an angle of elevation of 33°. The target H is 11 cm below the dragon's horizontal eye level. A human at point J looks up at the dragon at an angle of 40°. The length of the fire stream (side j = DH) is to be found. 2. **Given:** - Angle at D: $33^\circ$ - Angle at J: $40^\circ$ - Vertical distance below eye level: 11 cm (0.11 m) - Distance DJ (between dragon and human): 7.6 m 3. **Find:** Length of the fire stream $j = DH$. 4. **Step 1: Find angle at H** Sum of angles in triangle $DHJ$ is $180^\circ$: $$\angle H = 180^\circ - 33^\circ - 40^\circ = 107^\circ$$ 5. **Step 2: Use Law of Sines to find $DH$** Law of Sines formula: $$\frac{DH}{\sin(40^\circ)} = \frac{DJ}{\sin(107^\circ)}$$ Substitute known values: $$\frac{DH}{\sin(40^\circ)} = \frac{7.6}{\sin(107^\circ)}$$ 6. **Step 3: Solve for $DH$** $$DH = \frac{7.6 \times \sin(40^\circ)}{\sin(107^\circ)}$$ Calculate sines: $$\sin(40^\circ) \approx 0.6428, \quad \sin(107^\circ) \approx 0.9563$$ $$DH = \frac{7.6 \times 0.6428}{0.9563} = \frac{4.886}{0.9563} \approx 5.11\text{ m}$$ 7. **Step 4: Adjust for vertical offset** Since the target H is 11 cm (0.11 m) below the dragon's horizontal eye level, the actual length of the fire stream $j$ is: $$j = \sqrt{DH^2 + 0.11^2} = \sqrt{5.11^2 + 0.11^2} = \sqrt{26.12 + 0.0121} = \sqrt{26.1321} \approx 5.11\text{ m}$$ The vertical offset is very small, so the length remains approximately 5.1 m. **Final answer:** $$\boxed{j \approx 5.1\text{ meters}}$$