An scientist standing at the top of a mountain $4 \mathrm{mi}$ above sea level measures the angle of depression to the ocean horizon to be $1.87^{\circ}$. Use this information to approximate the radius of the Earth to the nearest mile. (Hint: The line of sight $\overline{A B}$ is tangent to the Earth and forms a right angle with the radius at the point of tangency.) The radius of the planet is approximately $\square$ mi. Check Save For Later Submit

Step 1 ：Given the height of the mountain above sea level is \(4 \mathrm{mi}\) and the angle of depression to the ocean horizon is \(1.87^{\circ}\)

Step 2 ：The angle of depression is equal to the angle between the radius of the Earth at the point of tangency and the line of sight from the top of the mountain to the horizon

Step 3 ：Using the tangent function, we have \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\), where the opposite side is the height of the mountain and the adjacent side is the radius of the Earth

Step 4 ：Rearrange the formula to solve for the radius: \(\text{radius} = \frac{\text{opposite}}{\tan(\theta)}\)

Step 5 ：Substitute the given values: \(\text{radius} = \frac{4}{\tan(1.87^{\circ})}\)

Step 6 ：Calculate the radius of the Earth using the tangent of the angle in radians

Step 7 ：The radius of the planet is approximately \(\boxed{123}\) mi.