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The answer is 32 pi.

omg thank you so much!

thank you so much!!

The possible values of k are 5 + sqrt(7) and 5 - sqrt(7), and their sum is 10.

The answer is [3,5].

There are 73 lattice points on the graph.

The area is 7 pi.

9                                  9

∫   2x    dx      =     x^2 ]      =     9^2  - 8^2   =    17

8                                 8

BTW....the "snake monster "   is known  as the "integral" sign   .....LOL!!!

Essentially.....we are evaluating the area between the line y  = 2x   and the x axis  from x = 8 to x = 9

This  is  a  trapezoid  with a height of  1...one base  = 16   and the other  = 18

So.....the area  becomes

(1/2)(height) (sum of the bases)  =

(1/2) (1) (16 + 18) =

(1/2) (34)  =

17.....just as we found with Calculus   !!!!

CPhill; The inadvertent typo of 180 should read 280.

Second one

The center  is    (h, k)  =  ( 6, -4)

a  = sqrt (36)  = 6       b   =  sqrt (49)   =  7

This hyperbola intersects the y axis

The  slope  of  the  asymptotes  =    ±√[36/49]  =   ±(6/7)

The equation of the asymptotes  is

y - k =  ±(a/b) ( x - h)    

so

y + 4 =  ±(6/7)(x - 6)

Here's a graph :  https://www.desmos.com/calculator/kalxu48bwq

I'll give you a hint for #2: this becomes \(\int_8^9x^2\).

You are very welcome!

:P

The  equation in the first problem is impossible....here's why...

If the center   is  (-2,4)  and a vertex is  (-2, 7)    then the major axis  lies  on the line  x  = -2...so this hyperbola will intersect the y axis

So "a"   =  3

So     the correct form  would  have to be

(y - 4)^2          (x + 2)^2

______     -    ________  =   1

    a^2                  b^2

The  positive slope  of  one  asymptote for this hyperbola   is  given by   a / b

So

3/b  = 1/2       

This implies that  "b"  = 6

Therefore....the correct  equation is

(y - 4)^2          ( x + 2)^2

_______   -    _________   =   1

     9                    36

Here is the graph :   https://www.desmos.com/calculator/a9eebyh8du

The numbers are 31, 58, 98, x, and x.
The mode, M, is x since it's repeated twice.
The mean, A, is 

a= (31+58+98+2x)/5

(187+2x)/5=3/2x

(187+2x)=15/2x

2(187+2x)=15x

374+4x=15x

11x=374

x=34



 

Cool! I can't confirm the answer, but it definitely makes sense.

Here's what I  get  as a "proof"

Let  "a" be the integer under the root....using  CU's quadratic, we have that

a  -  x  = x^2

x^2  + x  -  a  =  0

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

x^2 + x  + 1/4   =   a + 1/4

x^2  + x + 1/4  =  (4a + 1)  / 4

(x + 1/2)^2  =   (4a + 1) / 4         take the positive root

x  + 1/2  =    sqrt (4a + 1)  / 2

x =  sqrt  (4a + 1)  - 1

      _______________

                   2

Note   ...

If   a  = 2....then   this evaluates  to   1

If a  =  6.....then  this evaluates to  2

????

Very nice, CU....I saved this one to my "Watchlist"  !!!!!

Proof, please.

Here:

\(\sqrt{2-\sqrt{2-\sqrt{2-...}}}\)

Set it equal to x

\(\sqrt{2-\sqrt{2-\sqrt{2-...}}}=x\) equation 1

Square both sides

\(2-\sqrt{2-\sqrt{2-...}}=x^2\)equation 2

Because it goes on for infinity, it makes sense for us to susbtitute equation 1 into equation 2,

\(2-x=x^2\)

Now solve.

\(x^2+x-2=0\)

\(x=(-2,1)\)

So the answer is 1, as the answer can't be negative.

A trick to remember is this formula.

\(n-\sqrt{n-\sqrt{n-...}}=\frac{n}{2}\)

Does anyone know why it works? A proof?

THX, tomsun  !!!

nice

Here's another way

Call the other endpoint  (x, y)

So  we have that

4 + x                        5  +   y             

_____   =   1          _______   =  -2                multiply both sides by  2

    2                             2

4 +  x   =  2              5  +  y   =  - 4

subtract 4                    subtract 5

from both sides           from both sides

x  = -2                       y =  -9

good work guest. I was about to give the answer. 

another hint: If you are only given one endpoint and a midpoint, you know what the middle of the line segment is. Since the midpoint is half of what the line segment's length is, all you have to do is find the distance between the endpoint given and the midpoint, then add that coordinate to your midpoint and get your other endpoint.

If BAC  =  115°.....then  angle EAF   also  = 115°   [vertical angles ]

So....in quadrilateral  EAFH  angle  FHE  =  angle BHC  =

360  -  angle EAF  = angle AFH - angle AEH   =

360  - 115 -  90  - 90   =

180  -  115   =

65°

From 4 to 1   is -3    add another -3      for x = -2

From 5 to -2  is -7   add another -7              y = -9