CPhill

Let  (h, k)  be the center of the circle

And the distance  from this point to each  of the points  will be  equal  (the radius)

So  we have this equation

( h - - 7)^2  + (k - 0)^2  =  ( h  - 7)^2  + (k - 0 )^2

(h + 7)^2    =  ( h - 7)^2

h^2 + 14h + 49  =  h^2 - 14h + 49

14h  =  - 14 h

28h  = 0

h  = 0

And we can solve for k, thusly

(0 - 7)^2  + ( k -0)^2  =   ( 0  - - 3)^2  + (k - 4)^2

49  + k^2  =  9  + k^2 - 8k + 16

49  = 25  - 8k

8k  =  25  - 49

8k  = - 24

k  = -3

So......the center of the circle  is   ( 0, - 3)

And the radius  is  given by  √  [ (0 - 7)^2  + ( -3 - 0)^2 ]  =  √ [ 49 + 9] = √58

So... the equation  of  the circle is

(x - h)^2  + ( y - k)^2  = r^2

( x - 0)^2  +  ( y  - - 3)^2  = 58

x^2   +  ( y + 3)^2  = 58

Here is a pic :  https://www.desmos.com/calculator/m2ugumny98

Since the line  y  = -x  + 4  has a slope of -1, then the line containing CD has a negative reciprocal slope  = 1

Therefore....the  slope of this line is

(4 - 0)  / ( m - k)  = 1

4  =  m - k

4 + k  =  m

Now...the height of the trapezoid is  4

And the length of  one base  is  k - - 1  =  k + 1

And the length of the other base  is 4 + k

So....the area of the trapezoid can  be represented as

(1/2) height   [ length of base 1  +  length of base 2  ]   =  34

(1/2) 4  *   [ ( k + 1)  +  ( 4 + k)  ]   =34

2  *  [  2k + 5 ] = 34

2k + 5   = 17

2k  =12

k  = 6

Here  is a pic :

So...either 2, 3, 4, 5 or 6 people show up

And the least common multiple for  2,3,4, 5 , 6  =  60

So.....the least number he could bake with one left   is  60 + 1   = 61

Thanks, X² !!!....here's another approach.....

Let  SC  be the base of triangle RSC.....and let  its length  =  8 - x

The slope of the line segment  AC   =    (0 - 8) / (8 - 0)  =  -8/8  = -1

And the equation of the line containing this segment is   y = - x + 8  ⇒  y  = 8 - x

So...let the height of  triangle  RSC = SR   = 8 - x

And we have that the area of triangle RSC =

(1/2) (8 -x) (8 -x)  =  12.5     multiply both sides by 2

(8 - x) ( 8 - x)  = 25     simplify

x^2 - 16x + 64  = 25

x^2 - 16x + 39 =  0     factor

(x - 3) ( x - 13)  = 0

Setting both factors to 0  and solving for x produces

x = 13   (reject)   or

x  = 3   (accept)

So.... since R  lies on the line  y  = 8 - x, its coordinates are  ( 3, 5)

And the positive difference of its x and  y coordinates  =  5 - 3 =  2

If  C  is reflected over the y axis, the point  C'  is  (3, 2)

So....the distance from  C  to C'  is just the distance  from  (-3,2)  to (3,2)  =

  √ [ ( 3  -  -3)^2  +  ( 2  - 2 )^2 ]   =

√ [ 6^2  + 0^2  ] =

√ 36  =

6 units

What is the universal set???   [ Is it all the even positive integers < 50??  ]

The triangle formed will be an  8 -15 -17  "Pythagorean Triple" right triangle

If rotated about  the shorter leg....a cone with a height of 8  and a radius of 15  is formed....and its volume will be  =  pi ( 15)^2 * 8   / 3  = 600 pi units^3

If rotated about the longer leg....a cone with a height of 15 and a radius of 8 is fromed....and it  volume will be  =  pi (8)^2 * 15  / 3  = 320 pi  units^3

So.....rotating  the triangle about the shorter leg produces the max volume  of  600 pi  units^3

Since  B and  C are reflections of A over l and m....then the angle formed by the intersection of AB with l  = 90° ....call this intersection point, S....and the angle formed by the intersection of AC with m  is also 90°...and call this intersection point , T

So....we have quadrilateral   SATO  with angle  SAT  =  angle BAC

So....angle  SAT   =  angle BAC  = [360  - 90 -90  - 28]  = [180  - 28]    = 152°

a)  The distance ( in km ) that the ship is from the lighthouse at any time  after 1 PM is given  by :

h (d)   =  √ [ 8^2  +  (d)^2 ]   where d is the distance ( in km ) the ship has traveled since 1 PM

b )   d(t)  =  20t    where  t  is the time (in hours) since 1PM

c)  h° d  =  h[d(t) ]   =  √ [ 8^2 + (20t)^2 ]    which represents the total distance (in km) the ship is from the lighthouse at  any time  t  (in hours)  after 1PM

Since  AD  is a diameter, then ABD will form a right triangle with angle ABD  = 90°

So...angle ADB   = 40°

And we can find AB  as :

sin (40)  = AB  / 36

AB  = 36 * sin (40)  = 23.140353948732  ≈   7.3658 pi  units