1. **State the problem:** A satellite orbits 1000 km above Earth's surface. The angle at the satellite between the line to Earth's center and the line to a receiving dish is 27°. We need to find the time a signal traveling at 300,000,000 m/s takes to reach the dish. 2. **Identify given data:** - Radius of Earth, $R = 6370$ km - Satellite altitude above Earth, $h = 1000$ km - Distance from Earth's center to satellite, $r = R + h = 6370 + 1000 = 7370$ km - Angle at satellite, $\theta = 27^\circ$ - Signal speed, $v = 300,000,000$ m/s 3. **Find the distance from satellite to dish:** The satellite, Earth center, and dish form a triangle with angle $27^\circ$ at the satellite. Since the dish lies on Earth's surface, the distance from Earth's center to dish is $R = 6370$ km. We want the length of the side from satellite to dish, call it $d$. Using the Law of Cosines in triangle with sides $r$, $R$, and $d$ opposite angle $\theta$: $$d^2 = r^2 + R^2 - 2 r R \cos(\theta)$$ Substitute values: $$d^2 = 7370^2 + 6370^2 - 2 \times 7370 \times 6370 \times \cos(27^\circ)$$ Calculate $\cos(27^\circ)$: $$\cos(27^\circ) \approx 0.8910$$ Calculate each term: $$7370^2 = 54,316,900$$ $$6370^2 = 40,576,900$$ $$2 \times 7370 \times 6370 \times 0.8910 \approx 83,665,000$$ So: $$d^2 = 54,316,900 + 40,576,900 - 83,665,000 = 11,228,800$$ Take square root: $$d = \sqrt{11,228,800} \approx 3350.7 \text{ km}$$ 4. **Convert distance to meters:** $$d = 3350.7 \times 1000 = 3,350,700 \text{ m}$$ 5. **Calculate time taken for signal to travel distance $d$:** $$t = \frac{d}{v} = \frac{3,350,700}{300,000,000} = 0.011169 \text{ seconds}$$ 6. **Round to nearest thousandth:** $$t \approx 0.011 \text{ seconds}$$ **Final answer:** The signal takes approximately **0.011 seconds** to reach the dish.