Question

You fly a hot air balloon 3.5 miles above the ground. What is the measure of BD⌢, the portion of Earth that you can see? Round your answer to the nearest tenth. (Earth's radius is approximately 4000 miles.)

Answer 1

`\Delta s \approx 334.545\,m`

Explanation:

Let assume that Earth is a sphere, the following trigonometric diagram is constructed and presented below. The central angle is given by this inverse trigonometric equation:

`\theta = 2\cdot \cos^(-1) \left((r)/(r+h) \right)`

`\theta = 2\cdot \cos^(-1)\left((4000\,mi)/(4000\,mi+3.5\,mi) \right)`

`\theta = 4.792^(\circ)`

The distance of the portion of Earth that can be seen is:

`\Delta s = (4.792^(\circ))/(360^(\circ))\cdot (2\pi)\cdot (4000\,mi)`

`\Delta s \approx 334.545\,m`