1. **Problem Statement:** A dragon at point D breathes fire at a target H located 11 cm below its horizontal eye level, with an angle of elevation of 33° from D to H. A human at point J looks up at the dragon at an angle of 40°. We need to find the length of the steam of fire, which is side $j$ (the distance between points H and J). 2. **Understanding the Triangle:** We have triangle DHJ with angles and sides: - Angle at D: 33° (angle of elevation of fire) - Angle at J: 40° (angle human looks up at dragon) - Vertical drop from D to H: 11 cm (H is 11 cm below D's horizontal line) 3. **Find the third angle at H:** $$\text{Angle } H = 180^\circ - 33^\circ - 40^\circ = 107^\circ$$ 4. **Use the Law of Sines:** The Law of Sines states: $$\frac{j}{\sin 33^\circ} = \frac{h}{\sin 107^\circ} = \frac{d}{\sin 40^\circ}$$ 5. **Find side $h$ (distance from D to J):** Since H is 11 cm below D's horizontal line, and angle at D is 33°, the vertical drop relates to side $d$ (opposite angle 40°) and side $w$ (horizontal from D to H). We use the sine of 33° to find $d$: The vertical drop is opposite to angle 33°, so: $$11 = d \sin 33^\circ \Rightarrow d = \frac{11}{\sin 33^\circ}$$ 6. **Calculate $d$:** $$d = \frac{11}{\sin 33^\circ} = \frac{11}{0.5446} = 20.2 \text{ cm}$$ 7. **Use Law of Sines to find $j$:** $$\frac{j}{\sin 33^\circ} = \frac{d}{\sin 40^\circ} \Rightarrow j = d \frac{\sin 33^\circ}{\sin 40^\circ}$$ 8. **Calculate $j$:** $$j = 20.2 \times \frac{0.5446}{0.6428} = 20.2 \times 0.8477 = 17.1 \text{ cm}$$ **Final answer:** The length of the steam of fire $j$ is approximately **17.1 cm**.