CPhill

You did a pretty good job !!!!

I would shift  the graph to the right by pi/6    so we  get

y = -20cos (pi*x / 6  - pi/6) +  51

It  lines up a little better this way.......the  temps will not be exact for all months  because we don't have constant changes between them,  but  this  is about the  best you can  do with a periodic trig function

See the graph here with a few selected points   : https://www.desmos.com/calculator/zxwgqqtjzl

This triangle is  impossible because

9 +  6    ≠   15

We have  a 30 - 60  -90   right triangle

The side opposite the right angle is    2 /sqrt (3)   times the angle opposite the 60°  angle

So  AC  =  2 / sqrt (3) *  15 =   30/sqrt 3  =  (30/3)sqrt (3)  = 10sqrt 3

And the side oposite the 30° angle = AB   =  (1/2)*10sqrt (3)  =  5 sqrt 3

We have this property in a right triangle with an altitude drawn to  the right angle

AD / CD = CD / DB

So

4 / 6  = 6 / DB

4DB =  6*6

4DB   =36

DB   = 36 /  4  =   9

Note that when r changes by   -12, x changes by   -1

So....slope    change in y /  change in x =   -12 /  1  =  12

Note  that   when x = 0  ,  r  =  5.....so 5 is the  intercept

So  the equation is

r =  12x  +  5   

f(n + 1)  = 3f(n)

Means that every successive term is just  3 times  the previous one....so....

f(0) = 4

f(1) =  3 *4  = 12

f(2)  = 3*12  =36

f(3) =  3 * 36  = 108

I think we can  solve this with similar triangles

Call the side length of the  cube, s

The diagonal  across the cube's base =  s * sqrt (2)

1/2 of this     =  s / sqrt 2

We can construct a triangle with a base =  radius of cone -  s/sqrt 2   =   3  - s/sqrt 2

And the height of this triangle =  s

And the other triangle has a base of  the radius of the cone = 3  and a  height of 8

And these triangles wil be similar....so.....

3/ 8 =  (3 - s/sqrt 2) / s       cross-multiply

3s =  8 (3 - s /sqrt 2)  

3s + 8s / sqrt 2  =  24

s ( 3 + 8/sqrt 2)  = 24

s =  24  / (3 + 8/sqrt 2) ≈   2.77  

Volume of the cylinder =  pi * R^2 * h

So  the frustum's   volume =  (2/3)* pi * R^2 * h        (1)

Volume of a frustum =    (1/3) * pi * h *  [  R + r + Rr ]      (2)

Set  (1)   = (2)

(2/3)pi*R^2*h =   (1/3)pi * h * [ R + r + Rr ]         simplify

2R^2  =  R + r + Rr

2R^2 - R  =   r + Rr

R (2R - 1 ) =  r (1 + R)

R ( 2R - 1 ) / (1 + R)  = r

r / R =      R(2R - 1)  /   [ (1 + R) * R ] =

(2R - 1) / (R + 1)

We need  one more 3 and one more 5 to produce a perfect square

So.....

3 x 5 =   15

Sorry....I didn't see anyone typing.....!!!