Chapter 3.1, Problem 25E

Reminder Round all answers two decimals places unless otherwise indicated.

The Umbra of the Moon This is a continuation of Exercise 23. A total eclipse of the sun occurs when we are in the umbra of the moon. The size of the moon’s umbra on Earth’s surface is so small ${}^2$ that we can consider that the umbra reaches its apex at Earth’s center. The sun is $93,498,600$ miles away from Earth. What is the radius of the sun? (The actual radius is $434,994$ miles. Because of our simplified assumptions, you will get a slightly different, though relatively close, answer.)

23. 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 sum 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 fir inside Earth’s umbra? What celestial event occurs when this happens?