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We have the form

y = Asin (Bx + C) + D

A = the amplitude = 4

To compute "B"  we have

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

C = phase shift    (none)  so    C = 0

D = midline  =  1

So.......we have

y =  4 sin (  pi/2 * x ) + 1

We only have these to choose from   [ 2, 3, 5 , 7 ]

We have 4  choices for each digit.....so

4⁴ =     256   positive integers

Deleted.....

Similar to the others, here is the graph: 

Let's say that you want to compute n!, also known as n factorial. This expression would be equal to 1 * 2 * 3 * 4 * .......... * (n-1) * n.

For example, if you wanted to compute 4!, then you would have 1*2*3*4 = 24.

0! is regarded as an empty product, so it is 1. 

Another way of representing a factorial is \(\prod_{i=1}^{n} (i)\).

Factorials are often used in complemetary counting, probability, and statistics as such.

-24

I was unable to get a camera to work, so You'll have to read carfully.

1.Construct a trapezoid with the description of the problem.

2.Name the verticies starting from the top left side, going clockwise, A,B,C and D respectively.

3.Draw altitudes going down from A and B. Name the points where they intersect DC F and E, F is on the left.

4.Let side \(\overline{AB}\) be a, and \(\overline{DC}\) be b. \(\overline{FE}\) is a, and \(\overline{EC}\) = \(\overline{DF}\) = \(\frac{b-a}{2}\).

5.Let the intersection of diagonals be O.

6.\(\triangle DOC\) is a 45-45-90 triangle.

See if you can work it out from there.

Thanx, Chris.... those are the equations I used in my graph....but I got bogged down in the algabraic solution of them......  

In the first equation, multiply by \(-\sqrt[16]{5}+1\) to cancel out all of the nasty roots and you get \(\sqrt[16]{5}x + x = 1\). Simplifying further results in \(x = \sqrt[16]{5}-1\) .\(\)

Since we want \((x+1)^{48}\), we have \((\sqrt[16]{5})^{48}\), which equals \(5^3 = \boxed{125}\)

-24

Here is graph:

We have the form

y = Asin (Bx + C) + D

A = the amplitude =  2

The period =  4pi

Thus....in 2pi....there is (1/2) of a period.....so "B" =  (1/2)

There is no phase shift....so......C = 0

Since the midline is 2, then D =  2

So  the equation is 

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

Here:  https://www.desmos.com/calculator/o4e9ngaqie

Frequency  = 1 / period

  1/(4pi) = 1/period       Period = 4 pi      2pi / 4pi = 1/2

Amplitude = 2

Midline = 2    ( this means it is shifted UP 2 from its usual midline of y=0

Y intercept  0,2

2 sin (1/2x) +2  

Thanks, EP

Here's an algebraic solution  :

Let the number of problems Ben solved per day = P

Let the number days that Ben takes = D

So  Anne solved   6( D + 4)  problems  = 6D + 24  problems       (1)

And Ben solved PD   problems          (2)

And Claire solved ( P + 3) ( D - 2)  problems           (3)

So  using (2) and (3)

  PD   = (P + 3) (D - 2)

  PD =  PD + 3D - 2P - 6

  3D =  2P + 6     

  P =  [ 3D - 6 ] / 2          (4)

And subbing (4) into   (2)   and equating to (1) .....we have that

[ 3D - 6 ]   ( D) / 2   =  6D + 24

[ 3D - 6 ] * D  =    12D + 48

3D^2 - 6D =  12D + 48

3D^2 - 18D - 48 = 0

D^2 - 6D - 16  =    0       factor

(D - 8) ( D + 2)  = 0

Since D is positive.....then D = 8

So....the number of problems is 6(8) + 24   =   72

Thank you

Pls asap though

Before reading this answer, you might want to take a look at 

i will

oh ok

well i saw this written by another guy, its sum of two bases over 2..

maybe use similar triangles to prove? IDK!

read the ranger's apprentice series its soooo good

It says, that the exterior angle of a polygon with m sides is equal to a regular polygon with g sides. So,

\(\frac{360}{m} = \frac{180(g + 2)}{g}\)

\(360g = 180gm + 360m\)

\(2g = gm + 2m\)

Factoring out g:

\(2g - gm = 2m\)

\(g(2-m) = 2m\)

So therefore, \(g = \frac{2m}{2-m}\).

-24

Lol, thanks for trying to help me though. I really appreciate it :)

Heck yah!   There ya go.... we needed the 'a'        21000 ft         TwentyFour posted the solution......

sweet so will read

Ayn Rand    Atlas Shrugged           Is one of my favs...it is a long read!  

OMG!!! Thank you so much!!!!!!!!!

Here is the graph I used     x = days   y = rate