Melody

How come a fish is an animal and a person is an animal but a fish is not a person?

MWizzard,

I have been playing with it in Desmos and with Wolfram Alpha.

In Desmos for some strange reason, the left hand side th angle is taken to be in degrees and the answer is is given in radians, I do not understand why it has done this.

Generally speaking,    

\(\displaystyle \lim_{x \rightarrow \infty} \;xsin(\frac{a}{x})=a\)

If you know calculus I think this can be shown with L'Hopital's rule.  

I have no intuitive understanding of why this is so but if you know L'Hopital's rule I can show your that it is true.

\(\displaystyle \lim_{x \rightarrow \infty} \;xsin(\frac{a}{x})=a\\ =\displaystyle \lim_{x \rightarrow \infty} \;\frac{sin(\frac{a}{x})}{\frac{1}{x}}=\frac{0}{0}\\ \mbox{This is indeterminant and we can use L'Hopital's rule to find the answer}\\ =\displaystyle \lim_{x \rightarrow \infty} \;\frac{\frac{d}{dx}sin(\frac{a}{x})}{\frac{d}{dx}\frac{1}{x}}\\ =\displaystyle \lim_{x \rightarrow \infty} \;\frac{\frac{d}{dx}sin(ax^{-1})}{\frac{d}{dx}x^{-1}}\\ =\displaystyle \lim_{x \rightarrow \infty} \;\frac{(-ax^{-2}cos(ax^{-1}))}{-1x^{-2}}\\ =\displaystyle \lim_{x \rightarrow \infty} \;\;+acos(ax^{-1})\\ =+acos(0)\\ =\;a \)

خوب اگر شما به من بگویید چه شده است شما سوال، من ممکن است قادر به پاسخ      

Hi Max, it is nice to meet you :)

I will assume degrees

\(cos^25+cos^210+..... cos^245+........+cos^280+cos^285\\ =cos^25+cos^210+..... cos^245+........+sin^210+sin^25\\ =cos^25+sin^25+cos^210+sin^210+.........+cos^240+sin^240+cos^245\\ =1+1+1+1+1+1+1+1+cos^45\\ =8+\frac{1}{\sqrt2}\\ =8+\frac{\sqrt2}{2}\\ =\frac{16+\sqrt2}{2}\\ \)

Yes that is what I thought was most likely but I still have not spent the time on this question that I would like too.

Questions like this are always tricky.  :)

I am going to try and adjudicate.    :)

  Two workers, if they were working together, could finish a certain job in 12 days. If one of the workers does the first half of the job and then the other one does the second half, the job will take them 25 days. How long would it take each worker to do the entire job by himself?

Rom says     the workers will take 10days and 20 days to finish 1 job individually

Then if they did half each that would be 5 days and 10 days = 15 days to finish

That doesn't seem right ://

Guest says 20 days and 30 days then that would be 10+15=25 if they do half each.

Mmm if they work together then. Obviously the faster one does more of the job.

\((\frac{1\;job}{20\;days}+\frac{1\;job}{30\;days})*12days\\ =(\frac{1*3\;job}{20*3\;days}+\frac{1*2\;job}{30*2\;days})*12days\\ =(\frac{3\;job}{60\;days}+\frac{2\;job}{60\;days})*12days\\ =(\frac{5\;job}{60\;days})*12days\\ days\;\;cancel\\ =(\frac{5\;job}{60})*12\\ =(\frac{5\;job}{5})*1\\ =1 \;job\)

That is what I need to see.

I am going with our guests answer - sorry Rom.

I think you both did well.  I am going to think about this one some more.

I'd like another mathematician to confirm what I have written here please.

MWizzard has asked me to comment on his answer.  (I hope I got your gender right MWizzard       ) 

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You have said that x/0 is  infintiy but what if is negative then that could not be right.

Basically you simply cannot divide by zero - it is always undefined.

What you can say is :

Find the limit of     6/x      where is is positive but as x tends to 0 then the answer would be +infinity

you would write it like this

\(\displaystyle\lim_{x\rightarrow +0} \;\frac{6}{x}\;=\infty\\~\\~\\ and\\~\\ \displaystyle\lim_{x\rightarrow -0} \;\frac{6}{x}\;=-\infty\\\)

Hi Solveit,

I don't know, sorry ...

When I was very little I rarely made mistakes but I was so slow it was rediculous.

When I got older I had to learn to hurry up and then I made a lot more mistakes.

I am a bit dyslexic so that made it worse too.  

If you have time to check your answers try and check them a different way to how you did it in the first place.

If you go over the same working as before you are likely to overlook the same mistake....

ALWAYS check your answers if it is easy and/or if you have time.

ALWAYS look at your answer and consider if it is reasonable.  eg if you get a probability of 1.5 then you should know it is wrong and you should state that even if you do not have time to redo it.

Sorry, I really do not have anything useful to say.....

20/2=10

One jump 1.5 inches less = 9'10.5"

And the other 1.5 inches more = 10'1.5"

Subtract 1958 from BOTH sides