CPhill

Here : https://web2.0calc.com/questions/help-with-geometry_135

To make this problem easier, let's just shift the parabola up one unit so the we have the parabola  y =  x^2

The height  of our triangle = x^2  and the base =  2x

To have an area of 8  we need to  solve this

(1/2)* 2x  * x^2   = 8

x^3  =  8

x = 2

At x = 2 the base will be  2(2)  = 4  and the height will be  2^2 =  4  = y

And similarly   for and area of  64  we need  to  solve this

(1/2) * 2x  * x^2 =  64

x^4 =  64

x =  4

At x = 4, the base =  2(4) = 8   and the height = 4^2  = 16 = y

So....we  need to  shift everything down  by one  unit

y = "r"  will range from   (4-1) =  3  to  (16 -1)  =   15

So  r =   [ 3 , 15 ] 

Let the quantity in the large bag =  Q oz

Then in the bonus bag the quantity  = 1.2Q oz

Cost of large bag = $12

Cost of bonus bag = $8

Cost per oz in bonus bag  =    8  / (1.2)Q  

Cost per oz in large bag    =    12 / Q

So

[8 / (1.2 Q) ] / [ 12/ Q ] = 

8 / (1.2 Q)  *    Q / 12   =

8 / (12 * 1.2) = 

8 / 14.4  =

80 / 144  =

5 /  9

                                    C

                                   33

                            E                  D

                                    O

    78     A                    F                        B 39

Let O be the center of the incircle

ECOD  forms a  quadrilateral  with angles  OEC and OED both being right

So  angle  EOD  =  180 - angle C  = 180- 33  = 147° =  minor arc ED

And angle EFD is an inscribed angle inrercepting this arc

So its measure = 147 /  2   =   73.5°

(2 + x) / 2 =  -17                        (4 + y)  /  2 =   8

2 + x =  -17 * 2                          4 + y  = 2 * 8

2 + x = -34                                 4 + y =  16                           

x  =  -34 -2                                   y =  16 - 4

x =  -36                                         y =  12

x^2 - 4x + y^2 - 6y - 36 = 25       

x^2 - 4x + y^2 -6y  = 25 + 36       complete the square on x, y

x^2 - 4x + 4 + y^2 - 6y + 9  = 25 + 36 + 4 + 9

25 + 36 + 4 + 9 =  r^2

74   = r^2

sqrt (74 )   = r

x^2 + 6x + y^2 - 14y = -c         complete the square on x,y

x^2 + 6x + 9 + y^2 - 14y + 49  = -c + 9 + 49

To have a radius of 4

-c + 9 + 49   must  = 16

So

-c  =  16 - 9 - 49

-c = -24

c =  24

The number of cookies  before Mrs. Johnson ate any =  (3)*(4/3)  = 4

The number of cookies  before the class ate any =  4 / (1/6)  = 24  = what Mrs. Johnson brought

Proof

24  (5/6)  =  20   what the class ate

4 left

4( 3/4)  = 3 what Mrs. Johnson ate

Only three factors of 4  =   1, 2 , 4

P(factor of 4)  =  3  / 24    =    1 / 8

Here's the solution posted by Heureka  :

https://web2.0calc.com/questions/circle-is-the-incircle-of-triangle-abc-and-is-also_1