CPhill

sqrt ( 3^2 + 5^2 + 11^2)  = sqrt [ 155 ]  units  ≈  12.45  units

l =  21 cm

Circumference  of  cone =  (120/360)* 2pi (21) =  14 pi   cm

Radius of  cone  =   14 pi / 2pi  =   7   cm  = r

Volume of cone =   (1/3) pi r^2 * [sqrt ( l^2 - r^2 ) ]  = 

(1/3) pi (7^2) * [ sqrt ( 21^2 - 7^2) ]   ≈ 1015.94  cm^2

L =   sqrt [ 92^2  + (33-25)^2] =   sqrt [ 92^2 + 8^2] =  4sqrt [533]

R = 33

r =25

Surface area = pi L (R + r) + pi (R^2 + r^2)  = pi [ 4sqrt (533) ](58) + pi ( 33^2 + 25^2)  ≈ 22211.49 units^2

S1  =  pi * 33^2           S2  = pi * 25^2       H  = 92

Volume  =   H/3 * ( S1 + S2 +  sqrt  [ S1 * S2 ])  = 

92/3  *  ( pi (33^2 + 25^2) +  sqrt [ pi*33^2 * pi* 25^2 ] ) ≈ 244612.78 units^3

f(c)  = -1

This  means that some  value of c  = x   produces  -1

This  value is  x =  c =   -1

What we want  is the   intersection of the  circle with the  equation

(x -2)^2 + (y - 7)^2 =  25      (1)

with the line  y = 13 - x     (2)

Sub  (2)  into (1)

(x - 2)^2  +  (13 - x - 7)^2  =  25

(x - 2)^2  + (6 -x)^2   = 25

x^2 - 4x + 4 + x^2 -12x + 26 = 25

2x^2 -8x + 30  = 25

2x^2 -8x + 5  = 0

The  x values are

[ 8 + sqrt ( 8^2 - 4*2 *5) ] / 4   =   [ 8 + sqrt (24) ] / 4  =  2  +  sqrt (6) / 2 

And 2 - sqrt (6) / 2

The y values are   13 - [ 2 + sqrt (6)/2]  =   11 - sqrt (6)/2

And   13 - [2 - sqrt (6)/2] =    11 + sqrt (6) / 2

So   the points are

( 2 + sqrt (6)/2 , 11 - sqrt (6) /2)     and ( 2 -sqrt (6)/2 , 11 + sqrt (6) /2)

This  graph is  its own  inverse

So

f(x)  = f^-1 (x)

So

f^-1 (4)  =  f(4)   =   4

The slope of the line = 3    (y changes by 3 when x changes by 1)

The point (1,2) is on the line

The equation of the line  (except at x = 2) is     

y = 3 ( x -1) + 2

y = 3x - 1 =   f(x)

When x = 7      y = 20

So    f(7)  = 20

And  f (f(7)) =  f(20)

When x =20 , y =  59

So

f (f(7)) = f(20)  = 59

The sophomores  can  occupy  any of  4  positions

And the can  be arranged in 4! = 24 ways

And for each of these the sophomores can be arranged in 3!  = 6 ways

So

4 * 24 * 6  =   576  ways

1,2,3,4,5,6,6

Number of possible identifiable sequences = 

(Number of rolls) !  / (Number of any repeats ) !   =      7! / 2!  =    2520

f(-2)  means  that we are looking  for  the y value on the  graph  when  x =  -2

Can you find it  ???