CPhill

The total number  of subcommittees of three members that can be formed  is  C(13,3)  =  286

The total number of subcommittees  consisting of three Republicans  is  C(5, 3)   = 10

So....the probability  is

10  / 286   =       5  / 143

Let the side of the square  be, S

So

S^2  =  260     take the positive root

S   =  √260

The length of the diagonal   =  √2 * √260   =  √2 *√2* √130  =   2√130  miles

We want to find out where

(x^2) + 2   =  (x + 2)^2          simplify

x^2  + 2   =  x^2  + 4x  +  4

2  =  4x + 4

- 2  =  4x

-1  =  2x

x  = - 1/2

Just as CU  found    !!!

Suppose f(x) is an invertible function, and suppose that f(2)=f^{-1}(2)=4. What is the value of f(f(2))?

We know that   f (2)   =  4

So  f(f(2))   =   f(4)

If f^-1(2)  = 4      then    f(4)    =  2

So

f(f(2))  = f(4)   =   2 

1 cubic yard  =   3ft x 3 ft x 3 ft   =  27 cubic ft

So

3 cubic yards  =   3 (27 cubic ft)   =   81 cubic ft

The four largest  possible integers are

987643

987634

987463

987436

Only the last is a multiple of 4

By the Pythagorean Theorem

AC^2  = AB^2  + BC^2

(2x - 6)^2 =  90^2  + x^2

4x^2 - 24x + 36  =  8100 + x^2    rearrange as

3x^2  - 24x - 8064  = 0       divide through by 3

x^2  -  8x  - 2688  =  0       complete the square on x

x^2   - 8x  + 16  =  2688 + 16     

(x - 4)^2  =  2704        take the positive root

x - 4  =   52

x  =  56

So    BC  =  56     and  AC   =  2(56) - 6  =  106

So...since AD is an angle bisector  we have that

AB / BD  =  AC / DC

90 / BD  =  106 / DC

So

BD/ DC  =  90  / 106   

So...there are  90 + 106  =  196 equal parts  of BC....and   DC   is  106  of them

So    DC   =  56 ( 106 /196)  = 212/ 7

So  the area  of triangle  ADC  =  (1/2) (DC) ( AB)   =  (1/2) (212/7) * 90  ≈  1363 units^2  

Here's a pic :

Suppose that g(x)  =  f^{-1}(x). If g(-15)=0, g(0)=3, g(3)=9 and g(9)=20, what is f(f(9))?

f(x)  must be the inverse  of  f^-1(x)

And   g(x)  = f^-1(x)    

So...the points  ( -15,0)   (0, 3) and   ( 3,9)    are on  g(x)  and  f^-1(x)

Which means  that the points  ( 0, -15)    (3, 0 )  and  ( 9,3)   are  on f(x)

So

f(9)  = 3

And

f ( f (9))   =  f(3)   =   0

We can let A  = (0,0)

And B  = (10, 0)

Since [ OAB ] = 30....then  we can let O be (5, 6)

The distance between  O  and any vertex  is   sqrt  (5^2 + 6^2)  =  sqrt (61)  =  circumradius

So C  must be   ( 5, 6 + sqrt (61) )

See the image, here : 

f(8)  =  sqrt (8)

g(f(8))= g(sqrt (8))  =  8

Each function evaluation will  "flip-flop"  back and forth between these two values

So.....f(g(f(g(f(8)))))  =     sqrt (8)