CPhill

Thanks, Melody.....that was a LOT of work   !!!!

See the following image :

Let AC   = 8     BC  = 15      AB   =17

The area of this triangle   = (1/2) (8)(15)  =   60

The  altitude of  this triangle  can be found as

60    = (1/2)  17 * height

120 /17   = height  = CD  

This height will form the radius of two  cones

The height of the smaller  cone can be found as   sqrt  [ 8^2  - (120/17)^2  ] = 64/17  = 

The height of the larger cone can be found as   17 - 64/17  =  225/17

So   when   triangle  ACD  is rotated about hypotenuse  AB   its  volume is

(1/3) pi (120/17)^2 * (64/17)   =   [307200 / 4913 ] pi

And when  triangle BCD is rotated  about hypotenuse AB  its  volume id

(1/3)pi  ( 120/17)^2 (225/17)  = [1080000 / 4913 ]  pi

So...the total  volume  is  ( [  307200 +  1080000]  / 4913  )  pi  =   

[1387200 / 4913 ]  pi   cm^3 

No prob  !!!

Multiplying  through by e  we  get

e^(Ln2 ^(4/3) )  = 2^(4/3)           and we  can write

e ^ [ (4/3) Ln 2 ]=  2^(4/3)       take the Ln of both sides

Ln e ^[ (4/3) Ln 2 ]  =  Ln 2^(4/3)         and we  can write

[ (4/3) Ln 2 ] Ln e   =  (4/3) Ln 2          ..... Ln e  =1....so  we  can ignore this

(4/3) Ln 2  =  (4/3) Ln 2       !!!!

If no triplets  are on the team =  C(11,5)  =  462  possible teams

If one triplet is on the team  C (3,1) *  C( 11,4)  =  990  possible teams

If two triplets are on the team =  C(3,2) * C ( 11, 3)  = 495 possible teams

So    462 + 990  +  495 =  

1947   different teams

0 ≤  # apples ≤ 9

Number of diagonals in a polygon is given by :

n ( n-3)  / 2    where n  = the number of  sides

So

n ( n- 3)  / 2  =   2n     simplify

n (n -3)  =  4n

n^2  - 3n  = 4n

n^2 - 7n  =  0

n ( n - 7)  =  0

Setting each  factor to 0  and solving for  n  produces   

n  = 0      reject

n =  7

tan D   =  4sinD

sinD               4sin D

______  =                            multiply both sides  by  cos D

cos D

sin D  = 4sinDcosD     rearrange as

4sinDcosD  - sin D   =  0   factor as

sinD  ( 4cos D - 1)    =  0

So...set each factor to  0  and solve

sin D  = 0  →  D = 0°  or 180°   (reject)

4cos D - 1   =  0

cos D =  1/4

But... cos D  =  sin F   = 1/4

A

12                    20

B          16                       C

AB =  sqrt (20^2  - 16^2)   = sqrt ( 400 - 256)  = sqrt (144)  =  12

ΔABC ≈  ΔDEF

Since the triangles are similar, then sin F = sin C  =   12/20  =  3/5