CPhill

8 ft 3 in  =  12 * 8 + 3 =    99 in  =  width of wall

Center of  width  =  99/2 =  49.5 inches  from  the wall

The left edge of the picture  is at   (49.5) - (10/2)  = 49.5 -  5  =  44.5 in  from  the  wall  =

3 feet  8.5  in  =     3 + 8.5/12 ft   =   3 + 17/24  ft

Original area = S^2

New area = (1.6 S)^2  =  2.56 S^2

% increase =   (2.56  - 1) * 100% =   156%

Rearrange as

x^2 + 10x - 19 =  0

The sum of the roots =  -10 / 1 =  -10  =   a + b

The product of the roots =  -19/ 1=  -19 =  ab

(2a  -3) ( 4b - 6)   =

8ab - 12b - 12a + 18  =

8ab -12 ( a + b)  + 18  =

8(-19) - 12(-10)  + 18   =

-152 + 120 + 18  = 

-14

Area to  be covered =   2 (40*10  + 50*10)  - 50  =     1750 ft^2 

No. of   cans required =  ceiling [ 1750  / 300  ] =      6 cans

\( $\overline{AB} \perp \overline{BC}$, $\overline{DC} \perp \overline{BC}$,\)

AB = 9    DC =  5   BC =  12

We have trapezoid

         D      5       C

                         12

A       E              B

              9

Let DE  be perpendicular to  AB

And DE  =  BC  = 12    And  AE  =  AB - DC  = 9 - 5  = 4

ADE forms a right triangle

By the P Theorem 

AD  = sqrt [ DE^2 + AE^2 ] =   sqrt [ 12^2 - 4^2 ] = sqrt [ 144 - 16]  = sqrt (128)  = 8sqrt (2)

The perimieter =    5  + 12 + 9 + 8sqrt (2)  =   26 + 8sqrt (2)  cm

Rearrange as

x =   y^2  - 8y   +  13                complete the square on y

x = y^2  - 8y  + 16  + 13  - 16

x = ( y - 4)^2  - 3

(x + 3)  = (y - 4)^2

The  vertex is   ( -3, 4)

Here's the graph  : https://www.desmos.com/calculator/tksqqrj5dc

Call the sides of the rectangle  a,  b

The diagonal of the rectangle =  2r    =   10

And  

a^2 + b^2 =   100

The area of the circle =  25pi

So.....the area of the rectangle   =  (25/3) pi =   ab  →    b =  (25pi) / (3a) ...b^2  = 625 pi^2 / (9a^2)

So

a^2 +  625pi^2 / (9a^2)  = 100       multiply through by  a^2   and rearrange

a^4   - 100a^2  + (625/9)pi^2   =  0

a^4  -100a^2   = - (625pi^2)/9      complete the square on  a

a^4 -100a^2 + 2500  = -(625/9) pi^2 + 2500

(a^2 - 50)^2  = - (625/9)pi^2  + 2500         take the positive root

a^2  - 50  =  sqrt  [- (625/9)pi^2 + 2500 ]  ≈  42.6 cm

a^2 =  42.6 + 50   

a^2  ≈ 92.6

a =   sqrt (92.6)  ≈  9.6 cm

b = sqrt ( 100 - 92.6) ≈  2.7 cm

Perimeter of rectangle  =  2 ( a + b)  =  2 ( 9.6 + 2.7)  ≈  24.6 cm

Students            10         50        30      10

Score                 70         80       90       100

Median  = 80

Mean  =    [  70*10  + 50*80 + 30*90 + 10*100 ]  / [ 10 + 50 + 30 + 10 ]  =   84

abs  ( 80 - 84 )   =  abs (-4)   =   4

Set the equations  =   to find the x coordinates (in terms of a) of the intersections of these two functions

3x + a =   x^2 +  x          rearrange as

x^2 - 2x   =   a              complete the  square on x

x^2  -2x + 1 =  a + 1

(x  - 1)^2  = a + 1           take both roots

x -  1 =  ± √ (a + 1)

x = ± √ (a + 1)  + 1

Plugging these two x values into  the  equation of the  line we  have that

y =  3 ( sqrt (a + 1) + 1) + a        or      y =  3 ( -sqrt (a + 1) + 1) + a   

y =  3sqrt (a + 1) + a + 3                      y =  -3sqrt (a + 1) + a + 3

Using  the square of the distance formula we have that

( (sqrt (a + 1) + 1)  - ( -sqrt (a + 1) + 1) )^2  +  ( (3sqrt (a + 1) + a + 3) - (-3sqrt (a + 1) + a + 3))^2 = 30

Simplify

( 2 sqrt ( a + 1) )^2   +  ( 6 sqrt (a + 1 )^2     =  30

4 (a + 1)  + 36 (a + 1)  = 30

40a  + 40  =  30

40a =  -10

a =  -1/4

Here's a graph :

Points  A and B  are the intersections of  the functions.......the midpoint of the segment connecting A and B  = (1, 2.75)

A circle with this center and a diameter of sqrt (30) = radius of sqrt (30) /2   will pass through both points

This hyperbola opens sideways...so.....let the center  be    at ( x , 2)

"c"  =  x -  3       →   c^2  =  x^2 - 6x + 9

"a"  =  x  - 4       →   a^2 =   x^2  - 8x + 16

"b"  = sqrt  [ c^2 - a^2 ]  =   sqrt [  (x^2 - 6x + 9) - (x^2 - 8x + 16)  ]  =  sqrt [ 2x - 7]  

The slope is given as

b / a   =  sqrt 2

sqrt [ 2x  -7 ]  / [x - 4 ]   =  sqrt 2

sqrt [ 2x - 7 ]  =  sqrt 2 *  [ x -4]           square both sides

2x - 7  = 2  [ x^2 - 8x + 16 ]

2x^2 - 18x + 39  =  0

          18  +√ [ 18^2 - 4*2 * 39 ]               18 +√ [ 12 ]               9 + √ [ 3 ]

x =    ______________________  =    ___________  =       ___________  ≈   5.366

                      2 * 2                                         4                               2