Hello, epetrik!

1. A gambler sets up a game in which she puts four $10 bills and one $20 bill in one box.

A player randomly tales one of the bills and keep it.

To break even on average, how much must the gambler charge the player to play?

The player will draw: .$\displaystyle \begin{array}{c}\text{a \$10-bill }\frac{4}{5}\text{ of the time} \\ \\[-4mm]

\text{a \$20-bill }\frac{1}{5}\text{ of the time} \end{array}$

In five games, the gambler can expect to pay out $10, $10, $10, $10, $20

. . a total of $60 over 5 game.

That is, he can expect to pay out an average of $\displaystyle \frac{\$60}{5} \:=\:\$12$ per game.

To break even, the gambler should *charge* $12 per game.

2. Brittany pays $4.00 for a square piece of wood,

which she makes into a stop sign by cutting the corners off.

What is the cost of the wasted parts? A stop sign is tranditionally a regular *octagon**.* Code:

: - - - x - - - :
- * - - * * * * - - *
: | * * |
: | * * |
: * *
: * *
x * *
: * *
: * *
: | * x-2a * | a
: | * * |
- * - - * * * * - - *
: a : x-2a: a :

The square board is $\displaystyle x\text{-by-}x$ feet.

. . She paid $4 for $\displaystyle x^2$ square feet of wood.

Let the corner pieces be $\displaystyle a\text{-by-}a.$

Then the side of the octagon is $\displaystyle x-2a.$

At the lower-right, we have an isosceles right triangle

. . with equal sides $\displaystyle a$ and hypotenuse $\displaystyle x-2a$.

Pythagorus: .$\displaystyle (x-2a)^2 \:=\:a^2 + a^2 \quad\Rightarrow\quad 2x^2 - 4xa + x^2 \:=\:0$

Quadratic Formula: .$\displaystyle a \;=\;\frac{4x \pm\sqrt{8x^2}}{4} \;=\;\left(\frac{2\pm\sqrt{2}}{2}\right)x $

Since $\displaystyle a < x$, we have: .$\displaystyle a \:=\:\frac{2 - \sqrt{2}}{2}\,x$

The area of the four triangles is: .$\displaystyle A \;=\;2a^2 \;=\;2\bigg[\frac{2-\sqrt{2}}{2}\,x\bigg]^2 \;=\;(3-2\sqrt{2})x^2 $ square feet

The fraction of wasted wood is: .$\displaystyle \frac{(3-2\sqrt{2})x^2}{x^2} \:=\:3-2\sqrt{2} $

The cost of the wasted wood is: .$\displaystyle (3-\sqrt{2})(\$4) \;\approx\;\$0.68$