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Enjoy your time away! 

The terms progressively generated by the summation are

4        (4 + 9)       ( 4 + 9 + 16)      (4 + 9 + 16 + 25)     ( 4 + 9 + 16 + 25 + 36)     =

4           13               29                      54                              90          using the S of D, we have

        9             16               25                       36

               7               9                   11

                       2                 2

                                0

We have three non-zero differences so the polynomial will be a cubic in the form

ax^3 + bx^2 + cx + d

So......we have this system of equations

a(1)^3 +  b(1)^2 + c(1) + d = 4

a(2)^3 + b(2)^2  + c(2) + d  =  13

a(3)^3 + b(3)^2 + c(3) +  d =  29

a(4)^3 + b(4)^2 + c(4) +  d = 54

a  + b  +   c  +   d  =  4

8a + 4b + 2c + d = 13

27a + 9b + 3c + d = 29

64a + 16b + 4c + d = 54

Solving this system produces

a = 1/3     b = 3/2    c = 13/6   d =  0

So the polynomial is

(1/3)x^3 + (3/2)x^2 + (13/6) x

x  [ ( 2/6)x^2 + (9/6)x + (13/6) ]  

x [  2x^2 + 9x  + 13 ]

_________________

               6

You  can sub  n for x, YEEEEEET...same difference....!!!

Agree with tertre.....MAXIMUM t would be 69 tix at $ 61 each to clear $4200.....

...Note that Chris found the number of tix  which will maximize REVENUE @ 4225 (65 tix at $65 each)

Thank you so much! You guys are the best. See you again in a month or so ;)

You need  3:1  ts2  to m2r3 for a complete reaction.....

you have 50 ml of ts2   so you would need  16.67 ml of m2r3 for a complet reaction to occur.....you have 20-16.67 = 3.33 ml excess of m2r3 which will remain in solution         M and R  

Ugh, my Latex is not working well. 

We see that there is a price of \((70-(t-60))\) for each of the tickets, because there a price of 70 minus 1 for each ticket after the 60th ticket. So the total price is  \((130-t)(t)\)and we want this to be greater than \(4200\). Rewriting this, we see that \(t^2-130t+4200<0.\) This means that \((t-65)^2<25\), so \(-5 Thus \(60  , and the largest possible  \(t\) is  \(\boxed{69}.\).

Let x be the number of tickets above 60 = number of $1 decreases from $70

So

Revenue   =   Quantity * Price      .....so we have....

Revenue =  (60 + x) ( 70 - x)      simplify

R =  -x^2 + 10x + 4200

The number of $1 decreases that will  maximize revenue is given by

-10 / (2 * -1 ]  =    -10 / -2   =   5

So.....the number of bulk tickets that will maximize the revenue is given by  (60 + 5)   =  65 tickets =  t

You know EP, I’m pretty positive whoever posted that was an idiot.

He could be a smart-ass troll … or maybe he was measuring the ORP of THC, LSD, or 4-phosphoryloxy-N,N-dimethyltryptamine and contaminated himself.

The ORP registers as idiot, but it’s not an exact measure.

GA

Because of symmetry, the center of the circle will lie  at  (0, k)

Let the equation of the  circle be

x^2 + (y - k)^2 = r^2

x^2 + y^2 - 2ky + k^2 = r^2

Using implicit differentiation, the slope of any tangent line to the circle is given by

2x + 2yy' - 2ky'  = 0

x + yy'  - ky' = 0

y'  ( y - k)  =  -x

y' =     [ x ] / [k - y]

The slope of the tangent line to the parabola at any point =   d/dx (x^2) =     2x

So....equating these slopes, we have that

2x =  x / [ k - y]

2 = 1 / [ k - y]

k - y =    1/2

y = k -1/2      this is the y coordinate of the intersection of the parabola and the circle

So.....the difference in the height of the circle's center and the tangent point(s) is just the difference in their  y values

So

k - (k - 1/2)  =    1/2   unit

WOW i did 3r^2 + 6r^2 = 8r^2 LOL

thank you so much, i looked at it for about 3 times before posting as well... 

You are correct in your assumption.....

instead of  2 r^3 + 8r^2 + 13 r + 6      I get

                  2 r^3 + 9r^2 + 13r + 6  .......     So when you were expanding, you got something a little befuzzled.....look at it again....G'Luck !

Is there any more info given for this question.......like the size or equation for the circle?  I think there are MANY circles that would fit the bill.....do you need an answer in terms of the circle size/equation?

A little web searching found this:

( https://sciencing.com/how-to-convert-mv-to-ppm-12618513.html)

one millivolt is equal to 11.000000e-03 PPM.

Converting would then be     5 ppm/.011 ppm/mv = 454.54 mv          ( Not positive about this answer...but I know there is one out there! )

Thank you!

Thank you, everyone! 

thank you

Oh yes, they can!

Reported.

Ummmmmmmm, this is a math forum my dude...

Thanks Rom. 

I will look :)