AnExtremelyLongName

Plan:

Treat the star like a circle

In a circle counting problem, $n$ people can be seated in $(n-1)!$ ways.

Solve:

$(10-1)!=9!$

$\boxed{362880}$

Welcome to the forum! 

I calculated the perimeter of the dodecagon. The answer is not the sum of two dodecagons, but nice try Cal!

CPhill works 24 hours a day and solve 60 problems per hour, when will he get bored?

How many hours passed?

For visulation:

7 - 8 am, 8 - 9 am, 9 - 10 am, 10 - 11 am, 11 - 12 am, 12 - 1 pm, 1 - 2 pm.

7 hours

1. x + 7 * 2 = 10

2. x + 14 = 10

3. x = -4

-4 Celcius

Hello Guest! Make an account! 

Hypotenuse of big triangle: $\sqrt{13}$

This is just a compilation of a bunch of equations!

By Pythagorean Theorem:

(1) $(2-b)^2+(3-d)^2=b^2-a^2$

(2) $b^2-a^2=d^2-c^2$

(3) $d^2-c^2=\sqrt{13}-a-c$

By similarity where $s$ is the side length of the square:

(4)  $\frac{a}{b}=\frac{d}{s}$

(5) $\frac{2}{3}=\frac{2-b}{3-d}$

(6) $\frac{2}{3}=\frac{s}{c}$

(7) $\frac{2}{3}=\frac{a}{s}$

General:

(8) $a+c+s=\sqrt{13}$

y=3

then y=9

What was the change? Oh sure, it was multiplied by 3.

That means x must be multiplied by 3 too!

3 * 10 = 30

yay

Imagine two right triangles. They are similar because the sun shines at the same angle or the problem wouldn't make sense with the info given.

1. $\frac{\text{shadow}}{\text{height}}=\frac{\text{shadow}}{\text{height}}$

2. $\frac{1.8}{3}=\frac{12}{x}$

3. Solve for $x$

Super-hard-impossible bonus question:

Given the information in the problem.

If you were able to stand on top of the mail box and look STRAIGHT up into the sky, the sky can represent a 2D coordinate plane. Where would the Sun be located in (x,y)?

Consider everything you know (sorry physicists)!

Hello Guest!  I cannot correctly interpret the problem based on the format. I CAN give you a method to solve for one side length of the dodecagon, which no doubt can help you solve the problem yourself.

Interpret the problem

It's a pizza!
Look at ONE of the slices of the pizza. It has a vertex angle of 30 degress. (We calculate this by doing 180 - (360/12) = 150, then 180-150)

We know that the two long sides of the pizza is 1.

By law of cosines:

c² = a² + b² - 2ab*cos(C)

Plug in and solve:

c² = 1 + 1 - 2cos(30)

c² = 2 - 2cos(30)

c² = 1.691497100224