CPhill

It's clear  that the first digit  must  either  be  a 1 or a 2

If the first digit is a 1 , we have 6 possibilities for the last digit  [4,5,6,7,8 or 9]

So  there  are  10 * 10 *  6   = 600   4-digit numbers  of this type

If the first digit  is  a 2, we have  2 possibilities for the last digit  [8  or  9]

So there  are 10 * 10 * 2  =  200     4-digts  of this type

So.....we  have   600 + 200  =  800    4-digit integers

   For convenience  let there be 100 total students

Look at the following two-way table  :

               Finance      Non- Finance      Total

Female        12                  35                 47                               

Male             9                   44                 53

                 __________________________

                   21                   79                100

(a)  From the table, the probability that a finance major is  female =  12 / [ 12 + 9 ]  =  12/ 21  = 4/7

(b)  The probability  that a randomly selected female majors in finance = 12 / 47 

Let x  be  the lbs of the cheaper food

Then 50 - x  = the lbs of themore expensive food

We can solve this

.50 x  + 1.50 (50 - x)  =  1.25 * 50

.50x  + 75  - 1.50x  =  62.5     

-x  + 75  = 62.5     subtract  75 from both sides

-x = -12.5    divide both sides by -1

x=  12.5

So....the lbs of the more expensive food used=  50 - 12.5  = 37.5

3 of the sectors are the ones  that do not let the player move

So......these 3 sectors  occupy a total central angle of   (3/9) * 360  = 360/3    = 120°

 So   we can solve  this

Area  =  (1/2) radius^2  sin (120°)

14.6  = (1/2) radius^2  * √3  / 2

14.6  = (√3 /4) * radius^2     multiply both sides by  4/√3

14.6 * (4/√3)  = radius^2    take the positive square root  of both sides

5.8 in  =  radius  

Slope  of  given line  =  [ 0-1] / [ 5 - 0 ]   =  -1/5

And  because  (0, 1)  is on this line, the y intercept  = 1

The slope of the  perpedicular line is    5

So  

The equation  of this line is

y = 5x + 1

Here's a graph of both lines  :

https://www.desmos.com/calculator/64jk49lpwh

As a footnote.....note that  the "formula"  for  finding  the  unknown distance = 

l AX^2  - BX^2 l   =  l CX^2 - DX^2 l

And we could solve for any  of these  by  knowing the other three values

Here's  the  way to  solve  this

Let  E  be the foot of the  perpendicular  drawn  from X  to  AB

By  the  Pythagorean Theorem

AX^2  = AE^2  + XE^2     and

BX^2  =  BE^2   + XE^2         

1^2  = AE^2  + XE^2

7^2 = BE^2  + XE^2     subtract  the  first equation form the  second and we have that

48  = BE^2 - AE^2      (1)

Likewise......

Let F  be the  foot of the perpendicular drawn from X to  DC

And  note  that  AE  = DF

And BE =  CF

So....by the Pythagorean Theorem  we have that

DX^2  = DF^2  + XF^2

CX^2  = CF^2 + XF^2

DX^2  =  AE^2 + XF^2

CX^2  =  BE^2  + XF^2       

DX^2  = AE^2  + XF^2

8^2 =     BE^2  + XF^2      subtract  the first equation  from the second

64 - DX^2  =  BE^2 - AE^2    (2)

Equating (1) and (2)   we have that

64 - DX^2  =  48       rearrange  as

64 - 48  = DX^2

16  =  DX^2     take the positive  root

4  = DX

Just a Dragan found   !!!!

The  side  length is  12√3   ...... 1/2   of  this is the  radius of the  cone  6√3

The  height  of the  cone =  6√3 * √3 =   6 * 3  =  18

So....the volume  of this rotated  equilateral triangle will be a cone with a radius  of 6√3  and a height of  18

So  .....the  volume  is

(1/3) pi * radius^2  * height  =

(1/3) pi * (6√3)^2 ( 18) = 

6 * 108 pi  =

648 pi  units^3

y = 4x +  5  rearrange  as

-4x  + y  = 5    (1)

Note  that if we divide  the second equation  by  3     we  get that

-4x  + y  =  5/3       (2)

But notice  that   - 4x + y    couldn't  be  equal to   two different  values  ( 5  and 5/3)

So.....as EP found....we have no solutions   !!!!

Divide  the first  equation through by   -4  and we get that

x + y  = 0   which implies  that   y  = -x

Sub this into  the second equation  for  y   and we have that

-5x  - 3 (-x)  = -6

-5x  + 3x =  -6

-2x  = -6   divide both sides  by  -2

x  = 3  

And  

y =  -x  =  -3