All Answers 358083 Answers

It's basically about math, but often riddles and puzzles are presented to provide some light relief!

Drawing a graph is always a good idea with these sort of questions:

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Hi Melody

The full version of the sine rule is as I stated it.

It's actually derived using the circumscribing circle.

(Join the vertices of the triangle to the centre of the circle O. Then, angle BOC = 2*angle A, and from (half of) the isosceles triangle BOC, sin(A) = (a/2)/R, so a/sin(A) = 2R. Repeat for B and C.)

Tiggsy.

It is really good to see you Tiggsy :)

I probably just have a brain freeze but where did the 2R come from?

I mean it is not a part of the sine rule :/

Jeffes02, you REALLY need to learn to use the words  'thank you'

It is not difficult, practice in a mirror if you need to.

'Good job', sounds condescending. So don't use it.

The person who answered your question was not a 'random guest' he/she is a real person who allocated their time and knowledge to helping you. 

I know that you did use the word 'appreciate'. It was good to see that but your gratitude wording definitely needs to be worked upon.

You provide heaps of great answers on the forum, I have thanked you on a number of occasions and your efforts really are appreciated.

When you ask questions it is you who needs to show gratitude. Besides being polite it will also work in your favour as people will want to answer your questions in the furure. 

A surd is a commonly used term used in Australia.

It means an irrational number that is under a square root. 

Maybe it applies to other roots other than 2 but I do not recall seeing it ever being used that way.

Usually it is the irrational square root of a positive integer.

So for this question it makes no sense as the answer is rational.

\(\quad \int(e^{2x+1})e^{-x}dx\\ =\int(e^{2x+1-x})dx\\ =\int(e^{x+1})dx\\ =\int(e*e^{x})dx\\ =e\int(e^{x})dx\\ =e*e^x+c\\ =e^{x+1}+c\)

\(\quad \int(e^{2x}+1)e^{-x}dx\\ =\int (e^x+e^{-x})dx\\ =e^x-e^{-x}+c\)

Do you mean 

e^(2x+1)   or   e^(2x)+1     ?

Good job! I spent hours trying to solve this in wolfram only to get paid promotions slammed in my face, and a random guest is able to solve my answer clearer than it, appreciated for clearing my problem up :D

(You can use my method for writing math expressions if you prefer:)

1.Write the formula you want to put in this forum somewhere.

2.Copy/Paste it into Desmos on the left

Site link: https://www.desmos.com/calculator

In which it will turn the formula you entered into LaTeX (A kind of system to write math expressions.)

3.Choose the Sigma symbol (LaTeX) option on the tool bar above.

4.Copy/Paste your formula from Desmos into the LaTeX interface.

5.Press OK.

5.Done!

(See you having a hard time dealing with brackets and spacings :) )

Hi,I will have to use mod(x-5) for the expression  on the left hand side.To mean the modulus,or positive value of  x-5. So I can only write it like this ---  bear with me.        

1/2(x) +mod(x-5) = 3     solve for  x      Solution -

Move 1/2(x) to the right hand side so that we can square both sides,and then solve the quadratic.

thus

{mod(x-5)]^2 = [3-1/2(x)]^2

x^2-10x+25  = 9-3x +1/4(x^2)         re-arrange to get

3/4(x^2) -7x+16=0

now use the quadratic formula

x =[ 7 plus or minus sqrt(49 -4(3/4)(16)]  / (3/2)

which simplifies to

(2/3) (7) (plus or minus 1)

= plus or minus (14/3)

Don't know what it means, either. I fact-checked the wikipedia and it seems to either be the short term for irrational numbers, or the nth-root of numbers. But this question is answered anyways :D

(Source: https://en.wikipedia.org/wiki/Surd)     -Disambiguation Page

Here's another (trig based ) method.

The sine rule says that \(\displaystyle \frac{a}{\sin A} = \frac{b}{\sin B}=\frac{c}{\sin C} = 2R\),

where \(\displaystyle \text{R}\) is the radius of the circumscribing circle.

So, using the notation in Chris's diagram, \(\displaystyle \frac{a}{\sin A} = \frac{5}{4/5} = \frac{25}{4}=2R\),   so   \(\displaystyle R = \frac{25}{8} = 3\frac{1}{8}\).

Tiggsy.

Hmm...I have a question....What is a surd?

If I mean \(\frac{a}{b}(2x-12)=\frac{c}{d}\), I should write (a/b)*(2x - 12) = c/d.

If I mean \(\frac{a}{b(2x-12)}=\frac{c}{d}\), I should write a/(b(2x - 12)) = c/d.
This is not unreasonable.

a/b(2x - 12) = c/d  can mean both.

Or, since indices comes before division:

\(\frac{121^{-1}}{2}=\frac{1}{2*121}=\frac{1}{242}\)

Good work of yours! It is really easily for people to confuse others with inappropiate use of brackets.

Input: 121^-1/2 as a surd

Intepretation: Express the number \(121^{-\frac{1}{2}}\)

We apply the rule of exponentials:

\(a^\frac{n}{m}\)=\(\sqrt[m]{a^n}\)

And:

\(a^{-b}=\frac{1}{a^b}\)

Merge the two rules and we get:
\(a^{-\frac{n}{m}}=\frac{1}{\sqrt[m]{a^n}}\)

Now plug \(a=121,m=2,n=1\) into the equation above:

\(121^{-\frac{1}{2}}=\frac{1}{\sqrt[2]{121^1}}\)

Simplify:

\(=\frac{1}{\sqrt{121}}\)

\(=\frac{1}{11}\)

Done :D

That is much simpler, especially for younger students.

Thanks Chris :)

Solve for X.
a/b(2x - 12) = c/d

1.                                            on both sides

\(\frac{a}{b}(2x - 12) = \frac{c}{d}\)                 [\(\times\frac{b}{a}\)

\(2x-12=\frac{bc}{ad}\)                      [+12       

\(2x=\frac{bc}{ad}+12\)                      [ / 2

\(x=\frac{bc}{2ad}+6\) 

2.

\(\frac{a}{b(2x-12)}=\frac{c}{d}\)                        [\(\times (2x-12)\) 

\(\frac{a}{b}=\frac{c}{d}(2x-12)\)                 [\(\times\frac{d}{c}\)

\(\frac{ad}{bc}=2x-12\)                      [+12

\(\frac{ad}{bc}+12=2x\)                      [ / 2

\(\frac{ad}{2bc}+6=x\)

\(x=\frac{ad}{2bx}+6\)

Both can be meant. Use brackets!

  !

Intepretation: Approx for \(x\) in the function \(C(x)=\frac{8000}{x-50}\) when \(C(x)=1250\)

 Solution:

Solve for \(x\) when \(\frac{8000}{x-50}=1250\)

Multiple both sides by a factor of \(x-50\):

\(1250(x-50)=8000\)

Expand:

\(1250x-62500=8000\)

Move the numerical term to the right:

\(1250x=70500\)

Divide both sides by the greatest common divisor of \(1250\) and \(70500\) (Which is equal to \(250\))

\(5x=282\)

Divide both sides by \(5\):

\(x =\frac{282}{5}\)

\(=56.4 \approx 56\)

Done :D

(You accidently posted the formula for \(C(x)\) in your question two times, be careful.)

Input: Solve for X. a/b(2x - 12) = c/d

Intepretation: Solve for \(x\) in \(\frac{a}{b}(2x-12)=\frac{c}{d}\)

Solution:

Multiple both sides by \(bd\):

\(ad(2x-12)=bc\)

Expand:

\(2adx-12ad=bc\)

Move the numerical term to the right:

\(2adx=12ad+bc\)

Divide both sides by \(2\)
\(adx=6ad+\frac{bc}{2}\)

Finally, divide both sides by a factor of \(ad\):

\(x=6+\frac{bc}{2ad}\)

Done :)

Input: 3ax/5 - 4c = ax/5

Intepretation: \(\frac{3ax}{5}-4c=\frac{ax}{5}\)

Multiply by \(5\) on both sides:

\(3ax-20c=ax\)

Move the \(x\) term to the left, numeric ones to the right:

\(3ax-ax=20c\)

Simplify:

\(2ax=20c\)

Divide by \(2\) on both sides:

\(ax=10c\)

Finally, divide \(a\) on both sides:

\(x=\frac{10c}{a}\)

Done.

Input : a - c/x - a = m

Intepretation: \(a-\frac{c}{x}-a=m\)

Cancel the \(a\) out on the left side:

\(-\frac{c}{x}=m\)

Multiply both sides by \(x\):

\(-c=mx\)

Divide both sides by \(m\):

\(x=-\frac{c}{m}\)

Done :)

Just like all other calculators written in C++, such as this one, the maximum input/output number that such a script can handle is around \(10^{308}\), the limit of C++ integer values, if exceeded, digit overflowing occurs and the calculator will represent the number as infinity (i.e. the flipped \(8\) symbol).

Make sure your numerator & denominator is both less than \(308\) digits long in base-\(10\), if one of them is over this limit in your fraction, that one will appear as infinity, while the other number will be fine. If both of them are over the limit, it will simply show infinity/infinity. Be careful not to exceed the calculators limits!