CPhill

Remember that the bisected apex angle  sets up the following relationship

x / 4   =  2.25 / 3       cross-multiply

3x  =  4 * 2.25

3x  =  9     divide both sides by 3

x  =  3

The second one is similar

x + 8           2x - 5 

____     =   _____           cross-multiply

  10               14

14 ( x + 8)  =  10 (2x - 5)      simplify

14x + 112  =  20x  - 50     add/subtract  50, 14x  to/from  both sides

6x  =  162        divide both sides by 6

x  =  27

2x^2   + 8x  + 14   =  0        divide through by 2

x^2  + 4x  +  7   =  0        subtract 7 ffrom both sides

x^2  +  4x  =  -  7         complete the square on x

x^2  + 4x  + 4  =  - 7  + 4

(x + 2)^2  =  - 3       take the root of both sides

x + 2  =   ±√-3

x + 2  =  ±√3 i

x  =  -2  ±√3 i  ⇒  First answer

x^2 + x  - 12    factors  as   (x + 4) ( x - 3)

3x^2 + 11x  - 4 factors as  ( 3x - 1) (x + 4)

x^2 - 4  =  (x + 2) (x - 2)

So we have

   (x + 4) ( x - 3)

  ___________

     (x - 2)

_______________         =

   (3x - 1) ( x + 4)

  ____________

   (x + 2) ( x - 2)   

(x + 4) ( x - 3)           (x + 2) ( x - 2)

____________    *   ____________       =

      (x - 2)                 (3x - 1) (  x + 4)

(x + 4) ( x - 3) ( x + 2) ( x - 2)

______________________    =

   (x - 2) ( 3x - 1) ( x + 4)

(x - 3) ( x + 2)

___________     =

     3x - 1

x^2 - x - 6

________

    3x - 1

We have 9   ways to fill the first position  (1-9), 9 ways to fill the second position (every number 0-9 except the first one we chose) and 8 ways to fill the third position (every number 0-9 except the first two we chose)  =  9 * 9 * 8    =  648

5, 7

10, 11, 13, 14, 15, 17, 19

20, 21, 22, 23, 25, 26, 28, 29

30, 31, 33, 34, 35, 37, 38, 39

40, 41, 42, 43, 44, 45, 46, 47, 49

50, 51, 52, 53, 55, 56, 57, 58, 59

60, 61, 62, 63, 65, 66, 67, 68, 69

70, 71, 73, 74, 75, 76, 77, 78, 78

80, 82, 83, 84, 85, 86, 87, 88, 89

90, 91, 92, 93, 94, 95, 97, 98, 99, 100

I'll let you count them

Nice, EP  !!!!

Second one

The first four points form a parallelogram  with a width of 6   and a height of 4.....so its area  is

(6 * 4)   =  24 units^2

The last three points form a triangle with a base of 6  and a height of  2

So...its area is  (base * height) / 2  = 6 * 2  / 2    =    6 units^2

So.....the total area  is    (24 + 6)  =  30 units^2

Thanks, EP.....here's another approach  using the Law of Sines

x / sin 30  =  8 / sin 45

x sin (45)  =  8 sin 30

x (1/ √2 )  =  8  (1/2)

x / √2  =  4

x  = 4√2  ≈  5.66    ...   just as EP found  !!!

NVM.....EP has the correct answer!!!!

Volume of a cylinder  =   pi * r^2 * h

325 pi   =  pi * 5^2  * h      divide out pi

325  =  25 * h       divide both sides by  25

13 cm   =  h

For the second one...let's let the original dimensions   =  1  x 7

If the scale factor is 16:1   .....each dimension  in the actual house is 16 times as much...so the area is

7* 16   *   1 * 16  =

7 * 16^2    =  1792 ft^2

Last one

The diagonal is the hypotenuse of a right triangle and "12"  is the length of one of the legs......the other leg will represent the other dimension of the rectangle

So its length is

sqrt  ( 18^2  - 12^2)  =  sqrt (180)  

So   the perimeter is   2 (12 + sqrt (180)   )    =  50.83 in