All Answers 358090 Answers

Excellent job guys... 

Also Correct!

Correct!

Hello HeartStorm907

Before I start I will make it very clear that I have not looked at question or answer properly.

I will also say that I have NEVER said it is ok to post AoPS questions, especially when you are just looking for the answer.

Why do I think you are just after the answer?

Becasue you did not notice that originally key elements of the questin were missing.

AND more importantly

Because you just told CPhill his long answer was 'wrong'

It is easy to have a fundamentally correct answer but due to some small error somewhere the last line could be incorrect.

Did you look at CPhill's working? Or did you just plug in the last line and dismiss all he had done for you.

I never saw a thankyou anywhere. (for his time and effort) 

Nor did I see you ask for any clarification of his working.

It will  (theoretically) travel

12 / ( 1 - .7)  =   12 / (3/10) =  (12 * 10) / 3  =   40 ft

However....if we set the sum of an infinite geometric series to 40

12 [ 1 - .7^r ] / [ 1 - .7 ]   =  40

12 [ 1 - .7^r ] / .3  = 40

12 [ 1 - .7^r ] =  12

1 - .7^r  =  1

-.7^r  = 0         

Which means that the number of times  (r) that it takes to get the swing to  stop is not determinable......however, with friction involved it will eventually stop

Find the square of the largest real solution to the equation \((g(x))^2 - (f(x))^2 = 1.\)

Hello wiseowl!

\((g(x))^2 - (f(x))^2 = 1\\ \dfrac{1}{(x-\dfrac{1}{x})^2}-\dfrac{1}{(x+\dfrac{1}{x})^2}=1\)

\(\dfrac{1}{(\dfrac{x^2-1}{x})^2}-\dfrac{1}{(\dfrac{x^2+1}{x})^2}=1\)

\(\dfrac{x^2}{(x^2-1)^2}-\dfrac{x^2}{(x^2+1)^2}=1\\ \)

\(\dfrac{4x^4}{(x-1)^2(x+1)^2(x^2+1)^2}=1\)

Calculated by WolframAlpha.

\(x=-\sqrt{\sqrt{2}-1}\\ x=\sqrt{\sqrt{2}-1}\\ x=-\sqrt{1+\sqrt{2}}\\ x=\sqrt{1+\sqrt{2}}\\\)

\(\color{blue}x\in \{-1.55377,-0.64359,0.64359,1.55377\}\)

  !

B = benedict's original # of cars

S = sam's cars =  B-130 

Now give away some cars 

B will now have     B -  3/4 ( B-130 )        and S will have   (B-130) - .4 ( B)        These two equations are now equal 

   B - 3/4 (B-130 )    =   ( B-130) - .4B    solve for B

   ,25B +  97.5  =  .6B - 130 

    227.5  =  .35 B 

B=650 cars at first    

7 * 7 = 49, which is equal to 7.   

Therefore, the smallest whole number that is equal to seven times the sum of its digits is 7   

This conclusion should tell us all we need to know   

about what the OP considers logic.  SMH.  

_.

I interpreted this question as follows:

What is a?

Hello Guest!

\((x-5^2)^2+(y-6)^2=4^2\\ y=3x-1\\ (x-5^2)^2+(3x-1-6)^2=4^2\\ x^2-10x+25+9x^2-42x+49-16=0\\ 10x^2-52x+58=0\\ x^2-5.2+5.8=0\\ x_{1,2}=2.6\pm\sqrt{2.6^2-5.8}\\ x\in \{1.620,\color{blue}3.580\}\)

\(y\in \{3.861,\color{blue}9.739\}\\ P_2\ (3.580,9.739)\\ P_1\ (0,5)\\ a= \dfrac{y_2-y_1}{x_2-x_1}=\dfrac{9.739-5}{3.58-0}\\a=1.324\)

\(\color{blue}a\ is\ 1.324\)

With P3 (1.620,3.861) the further a can be calculated.

  !

Hmmmm....   I dunno....    to ME   digits   is plural...as in more than one digit....

   others may read this as you did which would make your answer correct....

So....this question is subject to interprotation.... making it a poorly worded problem !   

If x is between 1 and 10 then f(x) = -1

If x is between 11 and 1000 then f(x) = -2

Hence the sum is 10*(-1) + 990*(-2) =...

For the first one assuming that f(1) = 20-5 then im guessing the function is that when n increases the number of fives subtracted from 20 increases via the number of n (f(n)).

f(10) = -20-5-5-5-5-5-5-5-5-5-5 sooooo it has 5 tens 

4. f(1) =  - 6 as its a recursive function meaning you need to find the previous function number