CPhill

x^3  +  23x^2  + 3x  -  5  =  0

This isn't capable of being factored

There is a "formula" for solving a cubic, but it's messy

Here's a graph of the approximate solutions :    https://www.desmos.com/calculator/bqq37uwev1

The approximate solutions are   x ≈  -22.859 , -.543, .403

We have   12 + 5 + 9   =  26  people who liked the movie

The number in the 18-34 group   is 12

So....the probability  od 18-34   given that a person liked the movie is :

12 / 26   =     6 / 13  ≈   .46

Third  one

P ( previous work experience  and rated satisfactory )

The probability is just the product of the percentages along the apprpriate tree branches =

.75 * .85   =    .6375   =  63.75 %  ≈  64%

Second one

First part

P (  on first cube and 2 on second cube)  =   1/6  *  1/6 =  1/36

Second part

We have 11 cars and 3 are red

Peter  has  3/11 chance of getting a red one

Assuming that Peter gets a red one. there are 2 red cars left out of 10  

So...the probability that Richard gets  a red one is  2/10  = 1/5

So

P ( A and B)   =   3/11  *  1/5   =    3 / 55  ≈  .0545  ≈    5.5%

First one

The length of the square's side doesn't matter.....the probability is independent of this

The radius of the circle  = r

The side of the square  = 2r

The area of the circle =   pi * r^2    (1)

The area of the square  =  (side)^2  = ( 2r)^2  = 4r^2   (2)

So....the area  betweem the area of the square and the area of the circle  = (2) - (1)  =

4r^2  - pi r^2   =    r^2 [ 4 - pi ]

So....this area to the total area of the square is the probability  =

r^2  [ 4 - pi ]               4 - pi

___________  =      ______     ≈  .2146   ≈  21.5%

   4r^2                          4

laps/ minutes    =   ( minutes, laps)

(a) Slope of line =   [ 4.4 - 2.2 ] / [ 11 -  5.5 ]  =  2.2  / 5.5  =  2/5

(b)  She ran 2/5  of a lap every minute

(c)  In  30 min, she runs  30  * 2/5  =  60 / 5  =   12 laps

p   =  t    =   s

And these are complementary to these :

q  =  r   =   u

And   cos  (angle)  = sin ( complement to that angle )

So

cos p  =  sin  u

cos p  = sin  q

cos q  = sin t

cos t  = sin u

1/4    2/3     4/7      1/2

Get a common denominator  between   4, 3, 7  and 2 =    4 * 3 * 7 =  84

So 

1/4  (21 / 21)  = 21/84

2/3 (28 / 28)  =  56/84

4/7 ( 12/12)  =   48/84

1/2  ( 42/ 42)  =  42/84

So....in order  from smallest to largest

21/84     42/84   48/84   56/84  =

  1/4         1/2      4/7         2/3   

We  have this  arithmetic  series

10  +  12  + 14 + ......+  ???  =  220

The sum of an   arithmetic series is given  by :

S  =  (n / 2)  [ 2a  +  (n - 1) * d ]    

Where a  is the first term, n is the number of terms  [ and the number of rows in this case ] and  d  is the difference between terms

So we have

(n / 2)  [ 2(10)  + ( n - 1) * 2 ]   =  220        multiply both sides by 2

n [ 20 + 2n - 2 ]   =  440

n [ 18 + 2n ]  = 440

2n^2 + 18n - 440  =  0      divide through by 2

n^2 + 9n - 220  =  0       factor

(n + 20) ( n - 11)  =  0

Set eacn  factor to 0  and solve for n  and we get that

n = -20    [ reject ]    or

n  = 11    [ accept ]

So....11 rows

sqrt [ 4  +  sqrt (63)  / 2 ]  + sqrt [ 4 - sqrt (63 /2) ]

Square this

4 + sqrt (63/2)  +  2 sqrt [ 4  +  sqrt (63)  / 2 ]  * sqrt [ 4 - sqrt (63 /2) ] + 4 - sqrt(63/2)

[ 4 + sqrt (63 / 2)   +    2 sqrt  [ 16 - 63 / 4 ]  + 4 - sqrt (63/2)  ]   =

 8  + 2 sqrt [( 64 - 63) / 4  ]  =

8 + 2 sqrt ( 1/4)  =

8 + 2 (1/2)  =  

9

Since both  quantities under the original radical are positive, take the  positive square root  of 9  =

3