hectictar

Thank you Melody, that is a really cool keychain!!

Thank you so so much!!!!

Let's start by expanding both sides and combining like terms:

\(\begin{array}{rcl} \left(a+\frac{1}{b}\right)\left(b+\frac{1}{c}\right)\left(c+\frac{1}{a}\right)&=& \left(1+\frac{1}{a}\right)\left(1+\frac{1}{b}\right)\left(1+\frac{1}{c}\right)\\~\\ \left(ab+\frac{a}{c}+1+\frac{1}{bc}\right)\left( c+\frac{1}{a} \right)&=&\left(1+\frac{1}{b}+\frac{1}{a}+\frac{1}{ab} \right)\left( 1+\frac{1}{c}\right)\\~\\ abc+b+a+\frac{1}{c}+c+\frac{1}{a}+\frac{1}{b}+\frac{1}{abc}&=& 1+\frac{1}{c}+\frac{1}{b}+\frac{1}{bc}+\frac{1}{a}+\frac{1}{ac}+\frac{1}{ab}+\frac{1}{abc}\\~\\ abc+b+a+{\color{gray}\frac{1}{c}}+c+{\color{gray}\frac{1}{a}}+{\color{gray}\frac{1}{b}}+{\color{gray}\frac{1}{abc}}&=& 1+{\color{gray}\frac{1}{c}}+{\color{gray}\frac{1}{b}}+\frac{1}{bc}+{\color{gray}\frac{1}{a}}+\frac{1}{ac}+\frac{1}{ab}+{\color{gray}\frac{1}{abc}}\\~\\ abc+b+a+c&=&1+\frac{1}{bc}+\frac{1}{ac}+\frac{1}{ab}\\~\\ abc+a+b+c&=&1+\frac{a}{abc}+\frac{b}{abc}+\frac{c}{abc}\\~\\ abc+a+b+c&=&1+\frac{a+b+c}{abc} \end{array}\)

At this point, let's substitute   x   in for   a + b + c   and   13   in for   abc    so we want to solve this for  x:

\(\begin{array}{ccc} 13+x&=&1+\frac{x}{13}\\~\\ 169+13x&=&13+x\\~\\ 13x-x&=&13-169\\~\\ 12x&=&-156\\~\\ x&=&-13 \end{array}\)_

length of one side of park   =   √[ 4225 sq m ]  =   65 m

perimeter of park   =   4 * 65 m   =   260 m

length of each circuit  =   5% greater than 260 m   =   105% * 260 m   =   273 m

total length needed  =  4 * 273 m   =   1092 m

Ah, I did not think about using floor(log(x)) as the upper limit!! That's even better!

If you click Login, then mouse over right underneath the login button, a link which says "forgot your password?" in white text should appear.

Click that and fill out the info there.

That's really cool!! 

If you want to try it, here's another (similar, maybe easier) challenge:

Make a function that will return the sum of the digits. So for examples:

f(123) = 6

f(100) = 1

f(4532)  =  14

f(3) = 3

f(0) = 0

And let's also assume here that  n ≤ 32

That's some cool stuff!

Part A

The degree of the numerator  is equal to the degree of the denominator, so the horizontal asymptote is

y  =  a / 1     which we know is  -4

a / 1  = -4

a  =  -4

There is a vertical asymptote at  x =  3

So we know when  x = 3 ,    x + c  =  0

3 + c  =  0

c  =  -3

Now we know:     f(x)  =  (-4x + b)/(x - 3)

There is an x-intercept at  (1, 0)  so we know

f(1)  =  0

(-4(1) + b)/(1 - 3)  =  0

(-4 + b)/( -2 )  =  0

-4 + b  =  0

b  =  4

Altogether:     f(x)  =  (-4x + 4)/(x - 3)