CPhill

Volumes =  

Volume of outside cylinder - Volume of  inside cylinder  

First solid  volume

pi (3 - -2)^2 * 5  -  pi(0 - - 2)^2 * 5 =

pi [ 125  - 20]  =

105 pi

Second  solid volume

pi  (5 - -5)^2 * 3  -  pi (0 - -5)^2 * 3  =

pi [ 300 - 75]  =

225 pi

Difference in  volumes  =    (225 - 105 )  pi   =   120 pi

Let P  = (x , 6-x)

We can equate distances (squared)

(10 - x)^2  + ( -12 - (6-x)^2   =  ( 2 -x)^2  + (8 - (6-x) )^2

(x -10)^2  + ( x - 18)^2  = (x - 2)^2 +  ( x + 2)^2

x^2 -20x + 100  + x^2 -36x + 324  =  x^2 - 4x + 4  + x^2 + 4x + 4

2x^2  - 56x + 424 =  2x^2 + 8

416  =  56x

x = 416 / 56   =   52  /  7

y = 6 - (52/7)  =  -10/7

(x, y)  = (52/7 , -10/7)

[ABCD]  =  [ EFGH] - [EBA] - [ FCB] - [CGD] - [ ADH]

[ABCD ] = (5 *3) - (1/2) [ 1 *2  + 4*1  + 2* (1/2)  + 4.5 *1 ]  =

15  - (1/2)  [ 2 + 4 + 1 + 4.5 ]  =

15  - (1/2) [ 11.5 ]  =

15 - 11.5 / 2  =

[30 - 11.5 ]  / 2 =   

9.25

(3 , 19)

(19, 3)

(5, 17)

(17, 5)

Pretty easy

(b^2 - a^2) / ( b -a) =  2

(b + a) (b - a) / (b - a)  =  2

b + a  =  2

If I could figure this  out, I'd be qualified to fly the Space Shuttle.....

\(\lfloor t\rfloor = 3t + 4 - \lfloor 2t \rfloor \)

(blue function on graph)  = (green function on graph)

Interesting !!!! .....the functions  never intersect.....so.....no solutions  for t

60pi = pi * r^2 * (83/360)

60  / (83/360)  = r^2

r = sqrt  [ 360 * 60 / 83 ] ≈  16.13 

(2x + 7) (x - 4) =  -31 + jx

2x^2 - x - 28  = -31  + jx

2x^2 - (1 + j) + 3 =  0

To have only one solution

(1 + j)^2 - 4(2)(3)  = 0

(1 + j)^2  - 24  =  0 

(1 + j)^2  =  24           take both roots

j + 1 =  sqrt (24)              j + 1 = -sqrt (24)

j  + 1 = 2sqrt (6)             j + 1 = -2sqrt (6)

j = -1 + 2sqrt (6)             j =  -1 - 2sqrt (6)   

Since A is positive we can just multiply DS2011's  answer through by -1

x - 2y =  -4