Difference between revisions of "User talk:Bobthesmartypants/Sandbox"
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==physics problem== | ==physics problem== | ||
+ | |||
+ | <asy>unitsize(1cm); | ||
+ | draw((0,0)--(0,10)); | ||
+ | draw(Circle((1,1),1)); | ||
+ | dot((0,0)); | ||
+ | draw(9.5*dir(80)..9.5*dir(70)..9.5*dir(60),EndArrow());</asy> | ||
+ | |||
+ | <asy> | ||
+ | unitsize(1cm); | ||
+ | draw((0,0)--10*dir(60)); | ||
+ | draw(Circle((sqrt(3),1),1)); | ||
+ | dot((0,0)); | ||
+ | draw(9.5*dir(50)..9.5*dir(40)..9.5*dir(30),EndArrow()); | ||
+ | </asy> |
Revision as of 12:58, 27 April 2014
Contents
Bobthesmartypants's Sandbox
Solution 1
First, continue to hit
at
. Also continue
to hit
at
.
We have that . Because
, we have
.
Similarly, because , we have
.
Therefore, .
We also have that because
is a parallelogram, and
.
Therefore, . This means that
, so
.
Therefore, .
Solution 2
Note that is rational and
is not divisible by
nor
because
.
This means the decimal representation of is a repeating decimal.
Let us set as the block that repeats in the repeating decimal:
.
( written without the overline used to signify one number so won't confuse with notation for repeating decimal)
The fractional representation of this repeating decimal would be .
Taking the reciprocal of both sides you get .
Multiplying both sides by gives
.
Since we divide
on both sides of the equation to get
.
Because is not divisible by
(therefore
) since
and
is prime, it follows that
.
Picture 1
Picture 2
physics problem