1. **State the problem:** We need to find the furthest distance from the front of the stage (point D) where the lighting crew should place a mark so that any actor under 1.9 m tall can stand and remain fully lit. 2. **Understand the setup:** - Point A is 8 m above point D. - Point C is 1.2 m from D along the floor. - The light beam covers the area between points B and C on the floor. - We want to find a point between C and B where an actor of height 1.9 m can stand and still be fully lit. 3. **Key idea:** The light shines from A to the floor. The height of the light beam at any point on the floor between C and B decreases linearly from 8 m at C (since AC is vertical) to 0 m at B (since B is on the floor). 4. **Calculate the height of the light beam at any point x meters from C towards B:** - The length BC is 5.2 m. - At C (x=0), height is 8 m. - At B (x=5.2), height is 0 m. The height decreases linearly, so the height $h(x)$ at distance $x$ from C is: $$h(x) = 8 - \frac{8}{5.2} x$$ 5. **Find $x$ where height equals 1.9 m:** $$1.9 = 8 - \frac{8}{5.2} x$$ Rearranging: $$\frac{8}{5.2} x = 8 - 1.9 = 6.1$$ $$x = \frac{6.1 \times 5.2}{8}$$ 6. **Calculate $x$:** $$x = \frac{6.1 \times 5.2}{8} = \frac{31.72}{8} = 3.965$$ 7. **Find the distance from D:** Since C is 1.2 m from D, the mark should be placed at: $$1.2 + 3.965 = 5.165$$ **Final answer:** The lighting crew should place the mark approximately 5.17 m from the front of the stage (point D).