Final answer:
To find the radius of the Earth, we can use trigonometry and the information given. From a satellite 400 mi above the Earth, the angle formed by the vertical and the line of sight to the horizon is theta = 65.266°. By using the tangent function and rearranging the equation, we can calculate the radius of the Earth to be approximately 348 miles.
Step-by-step explanation:
To find the radius of the Earth, we can use trigonometry and the information given. From a satellite 400 mi above the Earth, the angle formed by the vertical and the line of sight to the horizon is theta = 65.266°. We can use the tangent function to relate this angle to the height of the satellite and the radius of the Earth.
First, convert the height of the satellite to kilometers: 400 mi × 1.609 km/mi = 643.6 km.
Next, we can use the tangent function: tan(theta) = R / (R + h), where R is the radius of the Earth and h is the height of the satellite. Rearrange the equation to solve for R: R = h / (tan(theta) - 1). Plugging in the values, we get R ≈ 643.6 km / (tan(65.266°) - 1). Using a calculator, we find that tan(65.266°) ≈ 2.146, so R ≈ 643.6 km / (2.146 - 1) ≈ 643.6 km / 1.146 ≈ 561 km.
Converting the result back to miles, we get approximately 561 km × 0.621 mi/km ≈ 348 miles. Since we are rounding to the nearest mile, the radius of the Earth is approximately 348 miles.