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hectictar

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Punkte9488
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 #5
avatar+9488 
+2

Well...I wanted to be consistent with the definitions recommended here  https://en.wikipedia.org/wiki/Gigabyte

 

I just really hope that in the future we can all agree on the same prefixes to mean the same things.

21.05.2019
 #3
avatar+9488 
+3

I have never heard of Appollonius's Theorem, but does seem like it can be used here.

 

Using the information from here:  https://en.wikipedia.org/wiki/Apollonius%27s_theorem

 

32 + 42  =  2( (2a)2 + a2 )  
25  =  2( 4a2 + a2 )

 

 

25  =  2( 5a2 )  
25  =  10a2

 

 

2.5  =  a2  
a  =  √[ 2.5 ]

 

 

BC  =  2a  =  2√[ 2.5 ]  
21.05.2019
 #1
avatar+9488 
+6

Let  a2=1644  and  b2=(2+5)211 , where  a  is a negative real number and  b  is a positive real number.

If  (a+b)3  can be expressed in the simplified form  xyz  where  xy,  and  z  are positive integers,

what is the value of the sum  xyzx+y+z ?

______________________________________

 

a=1644=4211=211 b=(2+5)211=2+511  (a+b)3=(211+2+511)3 (a+b)3=(2+2+511)3 (a+b)3=(511)3 (a+b)3=511511511 (a+b)3=551111 (a+b)3=5511111111 (a+b)3=555121

 

Now it is in the form  xyz  where  x,  y,  and  z  are positive integers.

 

x + y + z  =  5 + 55 + 121  =  181

21.05.2019
 #1
avatar+9488 
+3

 

By the Law of Sines:

 

sinB10=sin(π6)6 sinB=10sin(π6)6 sinB=56 B56.44°orB123.56°

 

Both options are valid in this case because neither make the current sum of the angles exceed  180° .

 

Using the first possible value of  B, that is,

B = arcsin(5/6)

 

Using the second possible value of  B, that is,

B = π - arcsin(5/6)

 

A=πBC A=πarcsin(56)π6 A=5π6arcsin(56) sin(A)=sin(5π6arcsin(56)) sin(A)=(12)(116)(32)(56) sin(A)=11+5312

 

By the Law of Sines:

 

sinABC=sinπ66 sinABC=112 BCsinA=121 BC=12sinA BC=12(53+1112) BC=53+11

A=πBC A=π(πarcsin(56))π6 A=arcsin(56)π6 sin(A)=sin(arcsin(56)π6) sin(A)=(56)(32)(116)(12) sin(A)=531112

 

By the Law of Sines:

 

sinABC=sinπ66 sinABC=112 BCsinA=121 BC=12sinA BC=12(531112) BC=5311

 

 

the first possible value of  BC  +  the second possible value of  BC  =  (53+11)+(5311)

 

the first possible value of  BC  +  the second possible value of  BC  =  103

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21.05.2019
 #4
avatar+9488 
+2
21.05.2019