geno3141

The graphing form is:   y - k  =  a(x - h)²   where the vertex is (h, k)  =  (2, 4).

--->   y - 4  =  a(x - 2)²  

To find a, substitute the values of (1, 1) into the above form:

--->   y - 4  =  a(x - 2)²   --->    1 - 4  =  a(1 - 2)²   --->   -3  =  a(-1)²   --->   -3  =  a

So, the equation is:  y - 4  =  -3(x - 2)² 

To get this into the desired form:  y - 4  =  -3(x² - 4x + 4)   --->   y - 4  =  -3x² + 12x - 12

--->   y²  =  -3x² + 12x - 8

r  =  6·sin(theta)  is a circle with radius 3 and center at (0,3).

You can use Heron's formula:  Area  =  sqrt( s · (s - a) · (s - b) · (s - c) )

where a, b, and c are the sides

and s is the semiperimeter = (a + b + c) / 2.

s = (5 + 7 + 8) / 2  =  10

Area  =  sqrt( 10 · (10 - 5) · (10 - 7) · (10 - 8) )

And you can finish ...

I agree with your answer of  5pi/3  but I think that the other answer should be  4pi/3.

(Drawing these in a unit circle both being the same distance below the x-axis.)

( 3x ) / [ x² + 2x + 1 ]  =  ( 3x ) / [ (x + 1)(x + 1) ]

( 2 ) / [ x² - 1 ]  =  ( 2 ) / [ (x + 1)(x - 1) ]

The lowest common denominator is:  (x + 1)(x + 1)(x - 1)

Multiplying the top fraction by  (x - 1) / (x - 1):

  ( 3x ) / [ (x + 1)(x + 1) ]  ·  (x - 1) / (x - 1)  =  [ (3x)(x - 1) ] / [ (x + 1)(x + 1)(x - 1) ]  =  [ 3x² - 3x ] / / [ (x + 1)(x + 1)(x - 1) ] 

Multiplying the bottom fraction by  (x + 1) / (x + 1)

  ( 2 ) / [ (x + 1)(x - 1) ] · (x + 1) / (x + 1)  =  [ (2)(x + 1) ] / [ (x + 1)(x - 1)(x + 1) ]  =   [ 2x + 2 ] / [ (x + 1)(x + 1)(x - 1) ] 

Subtracting the second from the first:  [  3x² - 5x - 2 ] / [ (x + 1)(x + 1)(x - 1) ] 

Area of a parallel is the base times the height.

When BC is the base, the height is DX.

So:  144  =  12 x DX   --->   DX  =  12

The points  (-8, 9)  and  (-2, 5)  are the endpoints of the diameter.

To find the center of the circle, find the midpoint between these two points:

              midpoint  =  ( (x₁ + x₂) / 2 , (y₁ + y₂) / 2 )  =  (  (-8 + -2) / 2. (9 + -5) / 2 )  =  (-5. 2)

To find the radius, find the distance from the center to one of the endpoints:

             radius  =  sqrt[ (x₂ - x₁)² + (y₂ - y₁)² ]  =  sqrt[ (-5 - -8)² + (2 - 9)² ]  =  sqrt( 9 + 49 )  =  sqrt(58)

The equation of a circle with center (h, k) and radius = r is:  (x - h)² + (y - k)²  =  r²  

             (x - -5)² + (y - 2)²  =  58

It will be a proportional relationship if:

1)  table of values:  use the slope formula -- if the slope between any two points remains the same.

2)  an equation:  if its a linear equation.

3)  a graph:  if the graph is any straight line that isn't either vertical or horizontal.

A cube with sides each 1/3 unit will have a volume of  (1/3)³  =  1/27  cubic units.

If you have a volume of  5  cubic units, then to find the equivalent amount of  cubes that each have a volume of 1/27 cubic units,

divide  5  by  1/27  =  5 x 27/1  =  135.

The 6s could be in positions 2 and 3:  066, 166, 266, 366, 466, 566, 666, 766

The 6s could be in positions 1 and 3:  606, 616, 626, 636, 646, 666, 676, 686, 696

The 6s could be in positions 1 and 2:  660, 661, 662, 663, 664, 665, 666, 667, 668, 669

Count them (just don't count repeats) and make certain that I didn't forget any ...