CPhill

The slope of the tangent line at  x  = 1  is   16(1) - 4   = 12

And at x  = 1, then  y  =  8(1) ^2  -4(1)  + 3   =  8 - 4 + 3  = 7

So...the equation of the tangent line is

y  = 12 (x - 1)  + 7

y = 12x  - 12 +  7

y  =12x - 5

Here's the graph : https://www.desmos.com/calculator/daj2o31drm

The horizontal tangent line will be tangent to the vertex.......it will have a slope of  0

So...we have that the x coordinate of the vertex is

16x - 4  = 0

16x  = 4

x  = 1/4

And when x  = 1/4, y  =  8(1/4)^2 - 4(1/4) + 3  =  1/2 - 1 + 3  =  2.5  = 5/2

So.....there is a horizontal tangent line at  ( 1/4, 5/2 )

We can make 7 different shakes if we use the same flavor

If we use different flavors, we can make   C (7 ,2)  =  21 diffferent shakes

So.......7  +  21   = 28 different shakes can be made

Using the Law of Sines

sin 69 / 35  = sin R / 30    multiply  both sides by 30

(30/35)sin69  = sin R

(6/7) sin 69   = sin R

Use the sine inverse to find R

arcsin [  (6/7) sin 69 ]  =  R  = 53°

We can use the Law of Sines,here

sin 61 / 45  = sin C  / 35      multiply both sides by 35

35 * sin 61 / 45  = sin C

(35 / 45) sin 61  = sin C

(7/9)sin 61  = sin C

Use the sine inverse to  find  C

arcsin  [ (7/9) sin 61 ]  = C  ≈   42.86° = 43°

A.

x^2  + y^2 - 4x -6y -12  = 0 

x^2  - 4x   + y^2   - 6y    = 12              complete the square on x and y

x^2  - 4x  + 4  + y^2  - 6y  + 9  =  12 + 4  + 9        factor the left side

(x -  2)^2  +  ( y - 3)^2   =  25

B.  This is a circle  centered at   ( 2, 3)   with a radius of 5

C. Here's the graph :  https://www.desmos.com/calculator/e4mzndmixp

3. A circle centered at the origin is tangent to the line x-y+4=0. What is the area of the circle?

Since the circle is centered at  (0,0)  a radius drawn  from this point will be perpendicular to this line at the point of tangency....and this will be the shortest distance from (0,0) to this line....so.....this distance  is given by

l  1 (0) -  1 (0)   + 4  l                       4          4√2             2√2     =  √8  units  = radius  of the circle

________________       =            ___  =    ____   =

   √  [1^2  + 1^2 l                            √2            2

So....the area  of the circle  = pi * radius ^2    =    pi  *  ( √8)^2   =   8 pi units^2

2.

5x-12y+45=0   and  5x-12y-46=0

12y   = 5x  + 45     and      12y  = 5x  - 46

y  = (5/12)x + 45/12

y = (5/12) x  +  15/4

These lines have the same slope.......the first line has  a  y intercept at  15/4....

So the point  (0, 15/4)  is on the first line

The distance between this point and the second line is the perpendicular distance between both lines...so we have

l  5(0)  - 12 (15/4) - 46  l               l  -12 (15/4)  - 46  l            l  -91  l              91       

____________________    =  ________________  =          ______     =      ___    =   7 units

      √ [ 5^2  + 12^2  ]                      √ [ 169 ]                             13                   13

1.

There may be several  ways to do this, but one of the easier ways  is through  the "formula" for finding the distance between a point and aline

If  (m, n)  is the point  and  Ax + BY + C = 0   is the equation of the line....the distance is given by

l  Am + Bn + C  l              l  4(3) + 3(4) + 7  l              l  31  l               31

______________  =  __________________  =    _______      =       ___  =  6.2    units

 √ [ A^2 + B^2 ]              √ [ 4^2 + 3^2  ]                    √  25                   5

If we let  f(x)  = y, we are looking for the the x coordinates of the form  ( x , 1.8)

The first segment which contains  a points like this   is the segment from  (0, 0)  to (1,2)

The slope of the line containing this segment  is  [ 2 - 0 ] / [1 - 0]    = 2

And the equation of this line is   y  = 2x

So...when y  = 1.8, we have

1.8  = 2 x     divide both sides by 2

.9  = x

So....the first point is  (.9, 1.8)

The next segment containing the y valu of 1.8  is  the segment from (1,2) to (2,1)

The slope of the line containing this segment is  (1 - 2)  /( 2 -1)  = -1/1  = -1

And the equation of this line is

y  = -(x -1) + 2

y  = -x + 3

And when y  = 1.8, we have

1.8  = -x + 3  subtract 3 from both sides

-1.2  = -x

1.2   = x

So.....the second point  is  ( 1.2, 1.8)

The last point where  y  = 1.8  is on the segment from (2,1)  to (3,3)

The sloe of the line containing this point is   (3 - 1)  /( 3 - 2)  =   2/ 1   = 2

And the equation of the line  containing this segment is

y =2 ( x -3) + 3

y = 2x  - 6 + 3

y = 2x  - 3

And when y  = 1.8, we have

1.8 = 2x  - 3    add 3 to both sides

4.8  = 2x   divide both sides by 2

2.4  = x

So....the last point is  ( 2.4, 1.8)

So....the sum of the x coordinates where  f(x)  = 1.8  is   .9 + 1.2 + 2.4  = 4.5

1.) degenerate hyperbola

2.) pair of intersecting lines