Chapter 3.1, Problem 23E

Reminder Round all answers to two decimals places unless otherwise indicated.

Earth’s Umbra Earth has a shadow in space, just as people do on a sunny day. The darkest part ${}^1$ of that shadow is a conical region in space known as the umbra. A representation of Earth’s umbra is shown in Figure $\mathrm{3.24.}$ Earth has radius of about $3960$ miles, and the umbra ends at a point about $860,000$ miles from Earth. The moon is about $239,000$ miles from Earth and has a radius of about $1100$ miles. Consider a point on the opposite side of Earth from the sun and at a distance from Earth equal to the moon’s distance from Earth. What is the radius of the umber at that point? Can the moon fit inside Earth’s umbra? What celestial event occurs when this happens?