CPhill

(a)

9 + 90*2  + 151 * 3  =  642  digits

(b)  Prob of 2   =

No. of 2's

In the ones place

10 in the first hundred

10 in the second hundred

5 in last 50

= 25

In the tens place

10 in the first hundred

10 in the second hundred

10 in the last 50

= 30

In the hundreds place

51  in  the last 51

Total 2's  = [ 25 + 30 + 51  ]  =  106

Prob of a 2  =  106  /  642  =    53  /  321

6 cubes have 1 black face

12 have 2 black faces

8 have 3 black faces

1 has no black faces (in the center)

Probability of  black face  =

(6/27) ( 1/6) + (12/27)(2/6) + ( 8/27) (3/6)  + (1/27) (0/6)   = 1/3

(The first set of parentheses contains the probability of that type of cube is chosen; the second set of parentheses contains the probability of having a black side facing up; mult/add these together for the total probability

Construct a circle centered at the origin with a radius of 1

The equation of this  circle  is

x^2 + y^2   = 1         (1)

Let P, Q lie on this circle

Line  through OP  has the equation

y = 4x

Sub this into (1)  to find the x coordinate of P

x^2 + (4x)^2   =1

17x^2  = 1

x^2 = 1/17

x =1/sqrt 17

y = 4/sqrt 17

P = (1/sqrt 17, 4/sqrt 17)

Similarly, line through OQ  has the equation

y =5x

Sub this into (1) to find the x coordinate of Q

x^2 + (5x)^2  =1

26x^2  = 1

x^2 = 1/26

x =1/sqrt 26

y = 5/sqrt 26

Q = ( 1/sqrt 26,  5/sqrt 26)

Slope of  PQ =   

[ 4/sqrt 17 - 5/sqrt 26 ] / [ 1/sqrt 17 - 1/sqrt 26 ]   =   [ 5 sqrt 26 - 20 ] / [sqrt 26 - 4 ]  =

[ 4sqrt 26 - 5sqrt 17 ] / [ sqrt 26 - sqrt 17 ]  =

[ 4sqrt 26 - 5sqrt 17 ] [ sqrt 26 + sqrt 17 ] / 9  =

[4 * 26  - sqrt (442) - 85 ]  / 9  =

[19 - sqrt 442 ]  /  9   ≈  -0.225

https://web2.0calc.com/questions/coordinates_80366

(p + 2q)  + (6p - 5q) + (14p + 8q)  =  180

21p + 5q  =  180

21p  =  [ 180 - 5q ]

p  =  [ 180 - 5q ] /  21

a^3 + 3ab^2  =  679       (1)

3a^3 - ab^2  =  615     →   9a^3 - 3ab^2  = 1845      (2)

Add (1) , (2)   and we  have

10a^3  =  2524

a^3  =  2524 / 10

a =    (252.4)^(1/3)

Sub this into   (1)

(2524/10) + 3 (252.4)^(1/3) b^2  = 679      solving this  for b produces

b = -4.74353  or b = 4.74353

a - b  = ( 252.4)^(1/3) + 4.74353  ≈   11.06

a - b =  (252.4)^(1/3) - 4.74353 ≈ 1.576

Simplify as

4x^2 + nx + 104 <  0

Set up as an equality

4x^2 + nx + 104 =  0

This will have real solutions when

n^2  - 4 (4) (104) ≥  0

n^2 ≥ 1664

n ≥ 40.79            or   n  ≤ -40.79

So....for our purposes   n < 40.79     and  n > -40.79

The number of  possible values for n  =  [ -40 , 40 ] =   81 possible values

x^2 + y^2   = r^2

(x + 2)^2 + ( y  -1)^2   = r^2            set these equal  for r^2

x^2 + y^2 =  x^2 + 4x +4 + y^2 - 2y + 1

4x - 2y  = -5     (1)

And

x^2 + y^2 = r^2

(x +3)^2 + ( y  -2)^2  = r^2           set these equal for r^2

x^2 + y^2  =  x^2 + 6x + 9 + y^2 - 4y + 4

6x - 4y =  -13      (2)

Multiply (1)  throughby -2  and add to (2)   and we have

-2x = -3

x =3/2

To find y, use (1)

4(3/2)  - 2y = -5

6 - 2y = -5

11 = 2y

y = 11/2

(x, y) = ( 3/2 , 11/ 2)

The vertex is  ( 3,1)  and the points  (0,7) and (6,7) are on the graph

c = 7

1 = a (3)^2  + b(3) + 7

7 = a(6)^2 + b(6) + 7            simplify

9a + 3b =  - 6    →  3a + b = -2 →   b = -2 - 3a     (1)

36a + 6b =  0       (2)

Sub (1)  into (2)

36 a + 6(-2-3a)  = 0

36a - 12 - 18a  =  0

18a =  12

a = 2/3

b = -2  - 3(2/3)  =  -2 - 2 =  -4

a * b * c =    (2/3) * (-4) * 7   =   -56 / 3

The axis of symmetry is 2

If   (4,7)   lies on the graph  then   (2 + 2 , 7)  =  (4, 7)

So

(2 - 2, 7)  =  (0, 7)  also lies  on the  graph

Sorry....see corrections

Simplify as

x^2 -6x + y^2 +10y  =  12 + 6               complete the square on x, y

x^2 - 6x + 9 + y^2 + 10y + 25 =  12  + 6+ 9 + 25

(x - 3)^2  + (y + 5)^2  =  52

Ths  is a  circle with  r^2  = 52

Area =   pi * r^2   =  52 pi

Law of Cosines

18^2  =  12.5^2 + PM^2 - 2 ( 12.5 * PM)  cos (QMP)

23^2  = 12.5^2 + PM^2 - 2(12.5 * PM)  (-cos (QMP)

Simplify

(18^2 - 12.5^2 - PM^2 ) /  [ -25 PM ]   = cos (QMP)

[23^2 - 12.5^2 - PM^2 ] / [ 25 PM =  cos (OMP)

Equate cosines  and simplify

- [ 167.75 - PM^2 ]   =  [372.75 - PM^2]

2PM^2  = 540.5

PM^2  = 270.25

PM =sqrt [ 270.25]  ≈  16.44

https://web2.0calc.com/questions/triangles_143