Hello, A Beautiful Mind!

The second one involves similar triangles.

9. A quarter (a 25-cent piece) is 3/4 inch in diameter and,

when placed 7 feet from the eye, will just block out the disc of the moon.

If the diameter of the moon is 2160 miles, how far is the moon from the earth? Code:

A
- *
: /|\
: / | \
: / | \
: / |84 \
d / | \
: B*-----+-----*C
: / ¾ \
: / \
: / \
-D*-------------------*E
: - - - 2160 - - - :

The eye is at $\displaystyle A.$

The coin is: .$\displaystyle BC = \tfrac{3}{4}$ inch.

The altitude of $\displaystyle \Delta ABC$ is 84 inches.

The diameter of the moon is: .$\displaystyle DE \:=\:2160$ miles.

The distance to the moon is $\displaystyle d$ miles.

Since $\displaystyle \Delta ABC \sim \Delta ADE$, we have: .$\displaystyle \frac{d}{2160\text{ miles}} \;=\;\frac{84\text{ inches}}{\frac{3}{4}\text{ inch}} $

Therefore: .$\displaystyle d \;=\;\frac{(2160)(64)}{\frac{3}{4}} \;=\;241,\!920$ miles.