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4.  

y  = Asin(Bx + C )  

A  =  the amplitude  =  [ difference in high/low/point ] / 2  =  6 / 2  =   3

The period is   20sec...so B  =

2pi / period  = 2pi / 20  = pi / 10

Since the sine graph starts at its highest position...the is a shift of pi/2 units to the left....thus  C  = pi/2

The function is :

y = 3sin ((pi/10) x   +  pi/2 )

The only question I have about this one is  at the beginning of the observation, x = 0 and the weight will be at its  highest point......thus.....the graph's first point will not be at the midline......that occurs 5 seconds later.... 

3.

In the form   y  = Asin (Bx)  + C

Period   =  4     we need to  find "B"  thusly

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

A  is the amplitude  =  4

C  is the shift of the midline from y  = 0.....thus.....the shift is 1 unit upward.....so C  = 1

The function is

y  = 4sin (pi * x / 2)  + 1

Here's the graph : 

2.

In the form Asin(Bx)  + C

Amplitude  = 2  = A

Midline  y  = 3  =  C

To find B   =   2pi/ period  =  2pi / 4pi   =  1/2

y intercept  (0, 3)

The function  is :

y  =  2sin( (1/2) x ) + 3

Note  that the parent function is :

y  = -2sin ( (1/2) x )  -  3

Here's the graph of both : https://www.desmos.com/calculator/xnb8aumkek

1.  Freq  =  1 / [ 6pi] 

The frequency  is  1 / period....so the period is 6pi....

In the form   y  = Asin (Bx)  + C

Amplitude  = 2  = A

Midline     .....y  = 3    =  C

To find B

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

y -intercept  =  (0, 3)

The function is : 

y  = 2sin ( (1/3) x)  + 3

Here's the graph :   https://www.desmos.com/calculator/8hza0yorxy

Perimeter =2[L + W]
Area =L*W

2[L + W] =22
L*W = 28, solve for L, W
L =7 and W = 4, or
W = 4 and L =7
 

I forgot to ask, can you show the steps you took to arrive at these answers?

f(x)  = 3 (x - 5)^2   + 9

1.  f(x) is shifted 6 units to right to obtain function g(x). What is the equation of g(x) in terms of f(x)?

g(x)  = f(x - 6)

2.  What is the equation of g(x) in terms of x?

g(x)  = 3 (x - 11)^2  +  9

3.  Then, g(x) is shifted 7 units down to obtain function j( x) . What is the equation of j(x) in terms of

f(x)?

j(x)  =  f(x - 6)  - 7

4.  What is the equation of j(x) in terms of x? 

j(x)  = 3 (x - 11)^2  + 2

EDIT to correct a previious omission....

For...the second.....the  bisected apex   angle   GEH sets up the following proportion :

GD / GE  =  DH / HE

x + 4  / 44.8   =  35 / 56    multiply both sides by 44.8

x + 4  =   44.8 * 35  / 56       simplify

x + 4  =  28      subtract 4 from both sides

x  =  24

Check for youself that  GD / GE  = DH / HE   =  5/8

Hope that helped......!!!!

The first and third one use the fact that  a line segment drawn parallel to a base splits the sides proportionally

So on the first

MA  =  46.2 - 14  =  32.2

And  we have the following proportion :

MA / MN  =  MB / MP

32.2 /  46.2  =  MB / 72.6       multiply both sides by 72.6

72.6 * 32.2 / 46.2  =  MB   =  x   =  50.6

For the third  one

5x + 10 / 10  =  121 / 11

5x + 10 / 10  =  11    multiply both sides by 10

5x + 10  = 110     subtract 10 from both sides

5x  = 100      divide both sides by 5

x  = 20     CORRECT!!!

There may be other ways of doing this....but....we can represent this as a piecewise function :

                  5.25x                       if    0 ≤ x ≤ 200

y   = 

                 5.25x  +  480           if      200 < x 800

The domain is  [0, 800]

The range  is [ 0, 4680 ]

Here's the graph :https://www.desmos.com/calculator/wfww9jpzwu

Thanks you guys! Super helpful!

Melody,

You don’t necessarily need expensive software to do Monte Carlo simulations (though it depends what you classify as “expensive”!).

You can do MC simulations using Excel (though I’m not a fan of Excel!).

I use an old version of Mathcad (unfortunately, PTC, who own Mathcad, now insist on an annual subscription for newer versions – and you can’t buy the older versions alone!).

I also use a home version of  MATLAB - price ~ £100 (100 UK pounds) – made by the Mathworks.

Let M  =  number of months in which deposits are made

Let T  be the total amount  in the account after M months  [ assuming no withdrawals ]

So...we have

T  = 723M  

And this is linear

Angle VTK  is bisected.....and a bisected angle sets up the following relationship

VY / VK  =  VT / TK   so we have

VY / 68  = 57 / 129.2       multiplyboth sides by 68

VY  =   68 * 57 / 129.2    =   30

So....  VY + YK =  x  =  30 + 68  =  98

Draw AD.....

This bisects  angle   BAC

And an apex angle that is bisected sets up the following relationship :

AB / AC  =   DB / DC

12/8  = DB /DC   (1)

But   DF  is parallel to base  CA....so  it splits sides  AB and AC proportionally

Thus

DB /DC  =  FB / FA

Subbing  (1)....we have that

12/8  = FB / FA

Thus.....side BA  is split into 20 equal parts......and  FA  =  AF  is 8 of these

So.... AF  =

(8/20) (12)  =  96/20  =   4.8

Deleted this answer....See Chris' answer !     Thanks CPhil !

12a = (m-n){a+1/2(m+n-1)(b-a)} , where m=13, n = 3

12a =(13 - 3)* [a + 1/2(13+3 -1)*(b -a)]

12a = 10*[a + 1/2(15)*(b - a)]

12a = 10*[a + 1/2(15b - 15a)]

12a = 10a + 5(15b - 15a)

12a = 10a + 75b - 75a

12a - 10a + 75a = 75b

77a = 75b

b =77a / 75, or

b/a = 77 / 75

= cisco          a networking hardware/software company

OH! Thanks so much!  I can use this actually for another problem that stumped me!

Move the decimal point 4 places to the left to get:

.000196   

1) b).

4 + 0.5t = 1.5 + 0.9t, solve for t

4 - 1.5 =0.9t - 0.5t

2.5 =0.4t    divide both sides by 0.4

t = 2.5 / 0.4

t =6.25 - years, when population of both towns will be the same

2,000 + 6.25 =2,006.25 =~2,006 - the year in which both towns will have the same population.

c) Sub t in one of the equations:

4 + 0.5t =

4 + 0.5 x 6.25 =

4 + 3.125 =

=7.125 millions, or:

7.125 x 1,000,000 =7,125,000 - the population of both towns when they will be the same in ~ 2006.

6t / t^-4 =6t x t^4 =6t^(1+4) =6t^5

m=5

n=-13

5m+n=12, 5n+m=5(n+1)=5*(-12)=-60

12 divides -60, but 5 doesn't divide -13