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We got the same result.....must be correct.....LOL!!!

Calculate all of the sq ft of the chute...

the top rectangle   8 * 12 = 96 ft^2

the rectangle below this one = 8 x 10 = 80 ft^2

and the two side triangles     2  *  1/2 x 10 x 6.5 = 65 ft^2

Added all together = 241 ft^2

241 ft^2  / .75 ft^2/person = 321.33 ~~~~321 persons

Just want to try this one....EPs answer might be better

We have that

[ m + 9 ] / 2 = x         [2m + 15] / 2 = y       [ 3m + 18 ] / 2 = z

So  the average of x,  y , z is

[ ( m + 9)  + (2m + 15) + (3m + 18) ] /  ( 2 * 3)  =

[ 6m + 42] / 6  =

m + 7

I come up with m+7

(x+y+z)/3 is the average....sub in the values given for x y and z

([9 +m   +2m+15  + 3m+18] /2   )/3  = ( (6m + 42)/2) / 3 = (3m+21)/3  = m+7

Add my best wishes.  Happy Birthday.  Does this count against my off-topic allowance?

HAPPY BIRTHDAY BRO 

Happy Birthday     !!!!!

u good bro

xy = 3/2      implies that   y = 3 / [ 2x]

So....plugging this into  the second function  and get that

10x + (3/5)(3/[2x]) =

10x + 9 / (10x)

If you haven't had Calculus....we can find the  minimum of this curve :  https://www.desmos.com/calculator/nbecr6spbo

The minimum value occurs at x = 3/10    and the minimum is  y = 6

If you have had Calculus....let the function be

y =  10x + (9/10)x^(-1)       take the derivative and set to 0

y ' =  10 - (9/10)x^(-2)  = 0

10  =    (9/10)x^(-2)    rearrange as

x^2 = 9/100       take the positive root   (since x must be positive)

x = 3/10

Taking the second derivative we have

y " =  (18/10)x^(-3)

Putting 3/10 into this produces a positive.....this indicates a minimum at x = 3/10

And y =  10(3/10) + (9/10)/ (3/10) =  

3 + (9/10) (10/3) =

3 + 9/3 =

6  !!!

Second one

A =  6

The period is 20 sec

B = 2pi /period =  2pi /20 =  pi/10

C = the normal midline is y = 0

The midline here is y = 10.....so C = 10

The function is 

y = 6sin [ (pi/10) x ] + 10

Here is the graph : https://www.desmos.com/calculator/pwtkr9csmq

CPhill is working on this...I know he does this differently than what I am going to show ya....

Definition of parabola is the set of points that are equidistant from a point called the focus and a line called the directrix.

This parabola has the focus BELOW the directrix, so it opens downward....it looks like a dome.

Point x,y on the parabola  has the same distance to the focus and the line y= 13

Distance to the focus is  (x-5)^2 +(y-2)^2  = d^2

Distance to the directrix is simply d = 13-y     d^2 = (13-y)^2       these two d^2  are equal

so

(x-5)^2 + (y-2)^2  = (13-y)^2         Expand and simplify to find your parabola equation

x^2 - 10x + 25    + y^2 - 4y +  =  y^2 -26y + 169        gather like terms and simplify and you will get

y = -1/22  x^2 + 10/22  x   + 140/22  for your parabola

                                              ...here is a graph..... Well...you can look at CPHill's graph !  

First one

We have the form    y = Asin (Bx)

A = 20/2 = 10

B = 2pi / period  = 2pi / 8   =  pi/ 4

C = midline =  y = 0

The function is

y = 10sin [ (pi/4] x  )

Here is the graph :  https://www.desmos.com/calculator/5dmvrecajh

\(A \begin{pmatrix}1\\ 2 \end{pmatrix} = \begin{pmatrix} 1 \\ 1\end{pmatrix}\\ 2A\begin{pmatrix}1\\ 2 \end{pmatrix} = \begin{pmatrix} 2 \\ 2\end{pmatrix}\\ A^{-1} \begin{pmatrix} 2 \\ 2\end{pmatrix} = 2 A^{-1}A\begin{pmatrix}1\\ 2 \end{pmatrix} =\begin{pmatrix}2\\4\end{pmatrix}\)

\(Av=\begin{pmatrix}1\\0\end{pmatrix}\\ A^{-1}Av = A^{-1}\begin{pmatrix}1\\0\end{pmatrix}\\ v = \begin{pmatrix}1&1\\2&x\end{pmatrix}\begin{pmatrix}1\\0\end{pmatrix}=\begin{pmatrix}1\\2\end{pmatrix}\)

(7,2)  and (2,12)

Find the slope = m  =   [ 12 -2] / [ 2 - 7 ] =  10 / -5  =  -2

Using either point we have - I'll use (7, 2) - we have

y = m (x - 7) + 2

 y = -2(x - 7) + 2      simplify

y = -2x + 14 + 2

y = -2x + 16

\(\text{two points }(7,2),~(2,12)\\ \text{the slope of a line segment joining them is given by}\\ m = \dfrac{2-12}{7-2} = \dfrac{-10}{5}=-2\\ \text{use point slope form with the first point}\\ (y-2) = -2(x-7)\\ y = -2x + 16 \)

We have the form

y = Asin (Bx) + C

A= the amplitude = 2

B =  2pi/period    = 2pi / pi   =  2

C = the normal midline is y = 0

y = - 2   shifts this down 2 units....so C  = -2

The function is

y = 2sin (2x) - 2      YOU WERE CORRECT!!!

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

The vertex is   ( 5, [13 + 2]/2 ) =  (5, 7.5)

p is the distance from the vertex to the focus = 7.5 - 2 = 5.5

The parabola turns downward because  the directrix is above the focus

We have the form :

4p ( y - k)^2 = - (x - h)^2         wher (h, k) = the vertex....so we have

4(5.5) (y - 7.5) = - (x - 5)^2      simplify

22(y - 7.5) = - ( x - 5)^2        divide both sides by 22

y - 7.5 = -(1/22)(x - 5)^2      add 7.5 to both sides

y = -(1/22)(x - 5)^2 + 7.5

Here is a graph : https://www.desmos.com/calculator/a8n6hmiw4w

A vertical asymptote means that a function is not defined at that x value

So....for instance.....if we have a rational function with vertical asymptotes of x = -2   and x =3....the domain is

(-inf, -2) u ( -2, 3)  U ( 3, infinity )

That is a 1 with 12431 zeroes after it.      10 to the 12431 power       10 x 10 x 10 x......   12431 tens....

Further info:

http://googology.wikia.com/wiki/Marioplex

A rational function is just that.....a ratio of functions

It may (or may not ) have a domain of all real numbers

It will have either a horizontal asymptote...a slant asymptote..or some other (more complex) asymptote

A polynomial is defined for all reals

There are no asymptotes

The range may (or may not) be all reals

Thanks for the help on the first one! Did you see my second question?

Here it is again in case you missed it.

Thanks!

Second one

We have the form   y = Asin (Bx) + C

A = the amplitude  = 2

Frequency =  1 /period  =  1 / (1/ 6pi)  =  6pi

B  =   2pi/ period =   2pi / 6pi    =  (1/3)

C = shift up/down

The normal midline is y = 0

y = 3  = shift up by 3 units.....so C = 3

The function is

y = 2sin ( 1/3 x) + 3

Here is the graph :  https://www.desmos.com/calculator/7erjhkrwuz