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Hi Melody: How would you approach this problem? Here is my thinking: A,B,C,D,E,F,G,H,I,J,K,L,M (13 letters). I'm thinking of leaving out the letter "A" and permute the other 12 letters as: 12 P 3 =1,320, 3-letter permutations. Or, should I compute 13 P 3 =1,716 3-letter permutations? Then simply append the letter "A" at the beginning or the end of each permutation, which would give 4-letter "words" and yet would NOT change the number of permutations. That would leave the number of permutations unchanged, whether it is 1,320 or 1,716, except that each permutation would now be a 4-letter "word" !

What do you think of this scheme? I value your opinion. Thanks.

Thank you!

Nvm I got it

As follows:

Given that f(3) = 5 and f(3x) = f(x) + 3 for all x, find f^{-1}(11).

f(3)  =f(3*1)=f(1)+3 = 5

so f(1)=2

f(3)=3+3 =5

f(9)=5+3=8

f(27)=8+3 =11

If   f(27) = 11       then     f^(-1)(11) = 27

Not exactly a derivation, but

It can be proved by induction.

Suppose that there is only one value of x  for which the distance from \(P_1(5,6) \)  to \(P_2(3x-7,ax+2) \)  is 4 . If  \(a \neq 0, \) what is  a?

Hello Guest!

\(d^2=(y_2-y_1)^2+(x_2-x_1)^2\)

\(4^2=(ax+2-6)^2+(3x-7-5)^2\\ 16=(ax-4)^2+(3x-12)^2\\ 16=a^2x^2-8ax+16+9x^2-72x+144\\ f(x)=(a^2+9)x^2-(8a+72)x+144=0\)

\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

\(x = {(8a+72) \pm \sqrt{{\color{blue} (8a+72)^2-4\cdot (a^2+9)\cdot 144}} \over 2\cdot (a^2+9)}\)    

When the radical is zero, the function f (x) has a single solution.

\( (8a+72)^2-4\cdot (a^2+9)\cdot 144=0 \)      | \(a \neq 0 \)

\(a=2.25\)

\(x = 3.2\)

  !

Thank you so much for your help!

There has been a request for further explanation on this one.

\(\frac{S_A}{S_B}=\left( \frac{R_A}{R_B} \right)^2\\~\\means\\~\\ \frac{Surface\:area\;of\:A}{Surface\:area\;of\:B}=\left( \frac{Radius\:of\:A}{Radius\:of\:B} \right)^2 \)

d = distance = rate * time

d is the same for each leg of the trip....out and back

60 * t1   =   40 * t2       divide both sides by 60

t1 = 2/3 t2

60 * t1   +    40 * t2   = a (t1 + t2)                      sub in    t1 = 2/3 t2

60 (2/3 t2)   +   40 t2   = a (2/3 t2 + t2)              divide both sides by   t2

40 + 40 = 5/3 a

a = 48 m/hr

You have to give either the distance between Washigton and Baltimore, or the time it took them to drive back and forth in order to calculate the average speed.

r = 1

x = 1 (mod 6)

x = 3 (mod 20)

x = 13 (mod 45)

x =180c  +  103, where c ==0, 1, 2, 3...........etc.

1,000,000 = 2^6 *  5^6

1,000,000 * (1 - 1/2) * (1 - 1/5) =

1,000,000 * (    1/2     *    4/5 )  =

1,000,000 * ( 0.4) ==400,000 - numbers that are relatively prime to 1,000,000.

N mod 10 ==5

N mod 13 ==6

N =130c  +  45, where c = 0, 1, 2, 3............etc.

45 mod 130 == 45

The slope between the given points is   [ 5 - 1 ] / [ 8  -1 ] =  4/7

The slope  of a perpendicular line to the one  through these two points is  -7/4

And  the midpoint between these two points  is  [ ( 8 + 1)/2 , (5 + 1)/2 ]  = ( 9/2, 3 )

So....the equation of  a line with a slope of -7/4  through the calculated  midpoint is

y = (-7/4) ( x - 9/2) +  3

y = (-7/4)x + 63/8 + 3

y = (-7/4)x + 87/8  =   the line of reflection

m = -7/4       b  =  87/8

m + b  =    -7/4 + 87/8   =     -14/8  + 87/8   =    73/8

Let's see if we can discover a pattern

[3, 5, 2, -3, -5, -2], 3, 5

The terms start  repeating after  the 6th term

And the sum  of  the 6  =  0

So...

2014 mod 6  = 4

Summing the first  2010  =  0

The reamaining terms  after summing the first 2010  is

3,  5 , 2 , -3    =    7

Call the volume  of  the original can     pi * r^2 * h

If the diameter is increased by 25 %  then so is the radius

So  we want to know this

pi * r^2 * h  =  pi *  (1.25 r)^2   (1 - a)  * h       where a is the % of height decrease

1  =  1.5625  (1 - a)

1/ 1.5625  =  ( 1 - a)  = 

.64  = 1 - a      rearrange as

a = 1 - .64  = .36   ......the height should be decreased by 36%

All numbers [8, 9, 10, 11, 12, 13, 14, 15]. Just sum them up.

Volume of water  held  when the tank is full

(1/3) pi * radius ^2 * height  =     (1/3) pi  * (16)^2 (96)  = 8192 pi   ft^3

When the tank is half- full.....the volume  =   8192 pi / 2   = 4096 pi ft^3

Using  silimlae triangles  we  have  that

r / h =  16 / 96 

r /  h =     1/6

r =   h/6

So.......we have  that

4096 pi  = (1/3) pi  r^2  h        subbing for  r  we have that

4096 pi  = (1/3) pi ( h/6)^2 * h        simplify

4096 = (1/3) h^3 / 36

4096 =  h^3  / 108

108 * 4096  = h^3

442368  = h^3        take the cube root of both sides

h ≈  76.2  ft

You have doubled counted.

Say the people are

A,B,C,D,E,F,G,H,J

Say you chose J as the single person then you chose  A,B,C,D for the first group and then you are left with E,F,G,H for the last group.

this is the same as

............Chose J as the single person then you chose  E,F,G,H for the first group and then you are left with A,B,C,D  for the last group.

I am pretty sure my answer is correct.

The "closed form" of the sequence is:

a_n = n^3 + 3 n^2 + 2 n 

Thanks Alan, that is a much better way to present it.  :)

90° clockwise rotation rule (about the origin)  for a point

(x , y)  →   (y, -x)

Swap x and y......change the  sign on x 

Can you take it  from here   ???

Volume = 6³ - 4/3 pi 3³

Looks good, hectictar  !!!!