+0  
 
+1
41
2
avatar

\(\sqrt{x^2+y^2}=100 \)

\(\sqrt{\left(x+\frac{4800}{\sqrt{937}}\right)^2+\left(y+\frac{3800}{\sqrt{937}}\right)^2}=299.289496902\)

Tried to write it in \(\LaTeX\) to be easier on the eyes.

Guest Oct 8, 2017
Sort: 

2+0 Answers

 #1
avatar+1224 
+1

This is indeed a messy problem. I'll solve for both variables, I guess.

 

1. Solve for a Variable

 

In this case, I will solve for x in the first equation, \(\sqrt{x^2+y^2}=100\)

 

\(\sqrt{x^2+y^2}=100\) The first step is to square both sides so that we eliminate the square root symbol.
\(x^2+y^2=10000\) Subtract \(y^2\) from both sides.
\(x^2=10000-y^2\) Take the square root from both sides to isolate x.
\(x=\sqrt{10000-y^2}\)  
   

 

2. Use Substitution to eliminate the x and solve for y

 

Plug in \(\sqrt{10000-y^2}\) for x into the 2nd equation and solve for y. This is not going to look good...

 

\(\sqrt{\left(x+\frac{4800}{\sqrt{937}}\right)^2+\left(y+\frac{3800}{\sqrt{937}}\right)^2}=299.289496902\) Plug in the value for x.
\(\sqrt{\left(\sqrt{10000-y^2}+\frac{4800}{\sqrt{937}}\right)^2+\left(y+\frac{3800}{\sqrt{937}}\right)^2}=299.289496902\) Square both sides to eliminate the square root.
\(\textcolor{blue}{\left(\sqrt{10000-y^2}+\frac{4800}{\sqrt{937}}\right)^2}+\left(y+\frac{3800}{\sqrt{937}}\right)^2=299.289496902^2\) In both binomials, we must follow the rule that \((a+b)^2=a^2+2ab+b^2\). I'll do the first binomial first, in blue.
\(\left(\sqrt{10000-y^2}+\frac{4800}{\sqrt{937}}\right)^2=\left(\sqrt{10000-y^2}\right)^2+2\sqrt{10000-y^2}*\frac{4800}{\sqrt{937}}+\left(\frac{4800}{\sqrt{937}}\right)^2\) Simplify this. Here we go...
\(10000-y^2+\frac{9600\sqrt{10000-y^2}}{\sqrt{937}}+\frac{23040000}{937}\) Now, let's remember that this equals the bit in blue. Now, let's expand the other part, in normal text.This time, I'll expand it without showing much.
   

 

\(\left(y+\frac{3800}{\sqrt{937}}\right)^2\) Expand this using the same technique as above.
\(y^2+\left(2y*\frac{3800}{\sqrt{937}}\right)+\left(\frac{3800}{\sqrt{937}}\right)^2\) Simplify further.
\(y^2+\frac{7600y}{\sqrt{937}}+\frac{14440000}{937}\)  
   

 

Time to do the simplification process. 

 

\(10000-y^2+\frac{9600\sqrt{10000-y^2}}{\sqrt{937}}+\frac{23040000}{937}+y^2+\frac{7600y}{\sqrt{937}}+\frac{14440000}{937}\) To make this easier to digest, I will rearrange the terms.
\(y^2-y^2+\frac{9600\sqrt{10000-y^2}}{\sqrt{937}}+\frac{7600y}{\sqrt{937}}+10000+\frac{23040000}{937}+\frac{14440000}{937}\) Now, let's do the simplification process.
\(\frac{9600\sqrt{10000-y^2}+7600y}{\sqrt{937}}+50000\) Great! Now that we have simplified as much as possible, let's reinsert this into the equation.
   

 

\(\frac{9600\sqrt{10000-y^2}+7600y}{\sqrt{937}}+50000=299.289496902^2\) We have to get rid of the remaining fraction by multiplying by the square root of 937 on all sides.
\(9600\sqrt{10000-y^2}+7600y+50000\sqrt{937}=299.289496902^2\sqrt{937}\) Subtract \(7600y+50000\sqrt{937}\)
\(9600\sqrt{10000-y^2}=229.289496902^2\sqrt{937}-7600y-50000\sqrt{937})\) Doing some more simplifying...
   

 

 

You know what...

 

This is really boring and tedious...

 

There are 2 ordered pair solutions to this. They are the following:

 

\(x≈68.53740254003346, y≈72.81912147963209\)

 

\(x≈86.60254037943613, y≈49.99999999828134\)

TheXSquaredFactor  Oct 8, 2017
 #2
avatar
+1

Sqrt(x^2 + y^2) =100

x =sqrt(10,000 - y^2)     Sub this into the second equation:

 

Solve for y:
sqrt((y + 3800/sqrt(937))^2 + (sqrt(10000 - y^2) + 4800/sqrt(937))^2) = 299.289

Raise both sides to the power of two:
(y + 3800/sqrt(937))^2 + (sqrt(10000 - y^2) + 4800/sqrt(937))^2 = 89574.2

Rewrite the left hand side by combining fractions. (y + 3800/sqrt(937))^2 + (sqrt(10000 - y^2) + 4800/sqrt(937))^2 = 400/937 (117125 + 19 sqrt(937) y + 24 sqrt(937) sqrt(10000 - y^2)):
400/937 (117125 + 19 sqrt(937) y + 24 sqrt(937) sqrt(10000 - y^2)) = 89574.2

Multiply both sides by 937/400:
117125 + 19 sqrt(937) y + 24 sqrt(937) sqrt(10000 - y^2) = 209828.

Subtract 19 sqrt(937) y + 117125 from both sides:
24 sqrt(937) sqrt(10000 - y^2) = 92702.6 - 19 sqrt(937) y

Raise both sides to the power of two:
539712 (10000 - y^2) = (92702.6 - 19 sqrt(937) y)^2

Expand out terms of the left hand side:
5397120000 - 539712 y^2 = (92702.6 - 19 sqrt(937) y)^2

Expand out terms of the right hand side:
5397120000 - 539712 y^2 = 338257 y^2 - 1.07831×10^8 y + 8.59377×10^9

Subtract 338257 y^2 - 1.07831×10^8 y + 8.59377×10^9 from both sides:
-877969 y^2 + 1.07831×10^8 y - 3.19665×10^9 = 0

Divide both sides by -877969:
y^2 - 122.819 y + 3640.96 = 0

Subtract 3640.96 from both sides:
y^2 - 122.819 y = -3640.96

Add 3771.13 to both sides:
y^2 - 122.819 y + 3771.13 = 130.178

Write the left hand side as a square:
(y - 61.4096)^2 = 130.178

Take the square root of both sides:
y - 61.4096 = 11.4096 or y - 61.4096 = -11.4096

Add 61.4096 to both sides:
y = 72.8191 or y - 61.4096 = -11.4096

Add 61.4096 to both sides:
y = 72.8191        or         y = 50.      or         x=68.5374     and           x=86.60254

Guest Oct 9, 2017

17 Online Users

avatar
avatar
We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details