Shoreline Illumination 2C681F

1. **State the problem:** We need to find the length of the shoreline illuminated by the spotlight from the Coast Guard ship. 2. **Understand the setup:** - The ship is $500$ m from the observer on the shore. - The spotlight illuminates up to $250$ m radius around the ship. - The observer's line of sight to the ship makes an angle of $20^\circ$ with the shoreline. 3. **Visualize the problem:** Imagine the shoreline as a straight line. The ship is located offshore, $500$ m away from the observer. The spotlight creates a circle of radius $250$ m around the ship. We want to find the length of the shoreline (line) that lies within this circle. 4. **Set up the coordinate system:** Place the observer at the origin $(0,0)$. The shoreline lies along the $x$-axis. The ship is located at point $S$ at distance $500$ m from the observer, at an angle of $20^\circ$ from the shoreline. Coordinates of the ship $S$: $$ S = (500 \cos 20^\circ, 500 \sin 20^\circ) $$ Calculate: $$ 500 \cos 20^\circ \approx 500 \times 0.9397 = 469.85 $$ $$ 500 \sin 20^\circ \approx 500 \times 0.3420 = 171.0 $$ So, $$ S = (469.85, 171.0) $$ 5. **Equation of the circle (spotlight range):** The circle centered at $S$ with radius $250$ m is: $$ (x - 469.85)^2 + (y - 171.0)^2 = 250^2 = 62500 $$ 6. **Find intersection points with shoreline:** The shoreline is the $x$-axis, so $y=0$. Substitute $y=0$ into the circle equation: $$ (x - 469.85)^2 + (0 - 171.0)^2 = 62500 $$ $$ (x - 469.85)^2 + 29241 = 62500 $$ $$ (x - 469.85)^2 = 62500 - 29241 = 33259 $$ $$ x - 469.85 = \pm \sqrt{33259} \approx \pm 182.33 $$ 7. **Calculate the $x$-coordinates of intersection points:** $$ x_1 = 469.85 - 182.33 = 287.52 $$ $$ x_2 = 469.85 + 182.33 = 652.18 $$ 8. **Length of shoreline illuminated:** The illuminated shoreline segment is between $x_1$ and $x_2$ on the $x$-axis. Length: $$ L = x_2 - x_1 = 652.18 - 287.52 = 364.66 \text{ meters} $$ **Final answer:** The length of shoreline illuminated by the spotlight is approximately **$365$ meters**.