CPhill

We know that, at t = 0, we have

6 = c - ( d - 4(0))^2

6 = c -(d)^2 

So...... c = 6 + d^2

And we also know that, at t = 2.5, we have

106  = c - (d -4(2.5) )^2    simplify

106  = c - (d - 10)^2

106  = c - d^2 + 20d - 100       and substituting for c, we have

106 = 6 + d^2 - d^2 + 20d - 100

100 = 20d - 100

200 = 20d       →  so  d = 10    and c = 6 + 10^2  = 106  

So.....at t = 1  we have

h(1)  = 106 - (10 - 4(1) )^2

h(1)  = 106 - (6)^2

h(1)  = 106 - 36    =   70 ft. when t = 1

Here's a graph :  https://www.desmos.com/calculator/f1v94eldv1

4x^2 - y^2 + 8x - 4y - 4 = 0

Complete the square on x and y

4[x^2 + 2x + 1]  -  [y^2 + 4y + 4]  =  4  + 4 -  4     factor and  simplify

4 ( x + 1)^2   - ( y + 2)^2    =  4       divide through by 4

(x + 1)^2  - (y + 2)^2 / 4  = 1      and we can put this in standard form as

(x + 1)^2  - (y + 2)^2   = 1

     1               4

The equation of the asymptotes is given by :

y = ± (b/a)(x - h) + k      and   b = 2,  a = 1,  h = -1    and k  = -2   so we have

y = ± (2/1) (x - -1) + (- 2)    which simplifies to

y = ± 2(x + 1) - 2

So the two two equations of the asymptotes further simplify to :

y = 2x   and y = -2x - 4

Here's the graph, Shades  : https://www.desmos.com/calculator/hguyqxzlql

Correct !!!

x^2 + 8x +19=0     

To complete the square.....take 1/2 of the coefficient on x  ......so (1/2)8   = 4

Square this  = 16

Add it and subtract it....so we have....

x^2 + 8x + 16 + 19  - 16   = 0     factor the first three terms

(x + 4)^2  + 19 - 16  = 0     simplify

(x + 4) ^2  +  3  = 0

That's it  !!!!.......you should verify that expanding this back out will give you the original equation....

This is going to be similar to the ellipse one, Shades.....just a few minor differences

Standard  form  looks similar to an ellipse except that a minus separates the first two terms instead of a plus

(x - h)^2      -   (y - k)^ 2      = 1

   a^2                 b^2

x^2 - y^2 - 2x - 4y - 4 = 0  we are gong to complete the square on x and y again.....this one is a little easier to do...   

x^2 - 2x  + 1   - [y^2 + 4y + 4]   =  4 + 1  - 4  simplfy

(x^2 - 1)^2    - ( y + 2)^2    =     1

Note that we can write this as  :

(x^2 - 1)^2    - ( y + 2)^2    =     1

     1                     1

And now, it's in S.F.

The center  is ( 1, -2)

The vertices lie at  (0, -2)   and (2, -2)

The equations of the asymptotes are, in this case, are given by :

y = ±(b/a) (x- h) + k     where a,b = 1    h = 1   and k  = -2   ....so we have....

y = ±(1/1) (x - 1) - 2     which simplifies to :

y = ± (x -1) - 2

Here's the graph with the asymptotes added :  https://www.desmos.com/calculator/i33hddox2i

Here's a pretty good resource for this one, as well

:http://www.purplemath.com/modules/hyperbola.htm

f(x)=9x^3 + 2x^2 - 5x+4 and g(x)= 5x^3 -7x +4

f(x)  - g(x)  =

                9x^3    + 2x^2   - 5x   + 4

         -  [   5x^3                  -7x   + 4 ]

                4x^3   + 2x^2   + 12x

x^2 + 2y^2 - 10x + 8y + 29 = 0

We need to put this in standard form which  is :

(x - h)^2      +  (y - k)^2       = 1

     a^2               b^2

Shades, we need to complete the square on x and y, first......rearranging and factoring out the lead coefficient on y^'2,, we have

x^2 - 10x + 25  + 2(y^2 + 4y + 4  ) =  -29 +  25 + 8   factor

(x - 5)^2   +  2(y + 2)^2   =  4       divide through by 4

( x - 5)^2   + (y + 2)^2    =  1

   4                    2

This is now in standard form

The center is ( 5 , -2)

The major axis lies along x.....  and its length is given by  2a   = 2(2)  = 4

The minor axis lies along y... and its length is given by  2b = 2(sqrt(2) ) = about 2.828

Here's a  graph : https://www.desmos.com/calculator/3yrngchn3l

Also...I wasn't able to go into great detail....but....here's something that might help you understand more thoroughlty..... http://www.purplemath.com/modules/ellipse.htm

blank / 2   = -3 / 6         cross-multiply

6 * blank  = 2 * -3        simplify

6 * blank  = -6            divide both sides by 6

blank  = -6/ 6

blank   =  -1

Stickers   Candy Pieces

      8            14

     16           28

     24           42

     32           56

     40           70

     48           84

     56           98

     68          112

     72          126

     80          140

     88          154

     96          168

    104         182

    112         196

    120         210

    128         224    

Volume of the sphere  =

(4/3)* pi  *(4)^3 =  about 268.08 cm^3

Volume of the cylinder =

pi *  (3)^2  * (5)  = about 141.37 cm^3

The sphere has almost twice the volume of the cylinder