James and Katie play a board game online which uses two dice to move around the board. Katie doesn’t have two dice, so uses a 12-sided spinner instead. The rules say that each player moves along the board the number of spaces shown on the two dice together. For example, throwing a 3 and a 6 means the player can move 9 spaces. An extra go is awarded each time the player rolls a double-six – James and Katie agree that Katie earns her extra ‘roll’ if she spins a ‘12’.After playing several games, James claims that Katie must be cheating because she gets many more extra ‘rolls’ than him on every game. Your Task:
• Try to decide whether James is right. Must Katie be cheating, or is there some other explanation for her luck?
• Use probability to explain why this problem has arisen
• Try to devise a fairer way to play the game.

let's add {f(x)=x+1}f(x)=x+1 and {g(x)=2x}g(x)=2x together to make a new function.

f(x)+g(x) =(x+1)+(2x)=x+1+2x=3x+1

let's call this new function hh. so we have:

{h(x)}={f(x)}+{g(x)}{=3x+1}h(x)=f(x)+g(x)=3x+1

we can also evaluate combined functions for particular inputs. let's evaluate function hh above for x=2x=2. below are two ways of doing this.

method 1: substitute x=2x=2 into the combined function hh.

h(x)

h(2)

​    

=3x+1

=3(2)+1

=7

​ since h(x)=f(x)+g(x)h(x)=f(x)+g(x), we can also find h(2)h(2) by finding f(2) +g(2)f(2)+g(2).

first, let's find f(2)f(2):

f(x)

f(2)

​    

=x+1

=2+1

=3

​  

now, let's find g(2)g(2):

g(x)

g(2)

​    

=2x

=2⋅2

=4

​  

so f(2)+g(2)=3+4=\greend7f(2)+g(2)=3+4=7.

notice that substituting x =2x=2 directly into function hh and finding f(2) + g(2)f(2)+g(2) gave us the same answer!

step-by-step explanation:

hans did not square the 40-unit side; he merely multiplied it by 2.   incorrect.

i got 10.28 but that rounds up to 10.3