CPhill

Law of Cosines

6^2  = 7^2  + 5^2  - 2(7)(5)cosACB

[ 6^2 - 7^2  - 5^2 ] / [ -2 (7)(5) ] =  cos ACB

-38 / -70  = cos ACB  =  19/35

sin ACB =  sqrt [ 1 - (19/35)^2 ]  =  sqrt  [ 1 - 361/1225]  = sqrt [ 864]/ 35 = 12sqrt(6) / 35  

1/r =  1/4 + 1/9 + 1/36 + 2sqrt [ 1/36 + 1/144 + 1/324 ]

1/r =  14/36  + 2sqrt [49/1296]

1/r = 14/36 + 2 (7/36)

1/r  = 14/36 + 14/36

1/r = 7/9

r = 9/7

(x-2) (x + 6) ≤ (x-2)(x + 5)

x^2 + 4x -12 ≤ x^2 + 3x -10

x ≤ 2

( -inf, 2 ]

An interior angle of  the n-gon  will measure

360 - 2(60) - 90  = 150°

So

(n-2) / n * 180 =  150

180n - 360 = 150n

30n = 360

n = 360 / 30   =   12

For convenience, let the volume of M = 20 and N =30    (M : N  = 20 : 30  = 2:3)

If M is is increased by 65%  it now holds   20 (1 + .65)  = 33

So

N must be increased by   ( [ 33 / 30 ]  - 1)  = 1.1 -   1 =  .1  =    10%

AD * AB  =180

The shaded area is a  trapezoid with bases of  AB,  AB/3    and  height AD

Its area =

(1/2)(AD)(AB + AB/3)  =

(1/2)(AD)(4/3)(AB)  =

(2/3) AD * AB  =     (2/3)(180) =  120

Let O be the center of the  circle

Angles  OTX and OUX = 90°

Angle TXU =  angle QXR = 60°

So   TOUX is   a quadrilateral  whose interior angles  sum to  360°

So angle   TOU  =   360 - 90 - 90 -60  = 120°  = minor arc  TU

And angle TAU is an inscribed  angle intercepting the  same  minor arc TU

So TAU measures  (1/2) (120)  = 60°

If all the psychology books are together, then so are all the mathematics textbooks

The books can be  arranged in two ways  (psychology first and  then math, or vice-versa)

And for each of these the math books can be arranged in 4! = 24 ways

And the psychology books can be arranged in 3! = 6 ways

So

6 * 24 * 6  = 864 arrangements

The sophomores can occupy  any one of 5 positions

For each of these they can  be  arranged in 3! = 6 ways

And for each of these the  freshmen can be  arranged in 4! = 24 ways

So

5 * 6 * 24  =   720 arrangements

(23, 29, 31, 37, 41, 43, 47)

He could choose any  pair except the last two

So

C(7,2)  - 1  =   20  different  products