CPhill

We  can fing this using some trig

We have a right triangle  with one leg  = y/2

And the angle   opposite this side = 22.5°

And   1/2 of the distance between the parallel sides is the other leg....and the angle opposite this side   is  67.5°

So....using the Law of Sines....we can find half of the distance as

(y/2)sin (67.5)  / sin (22.5)  =  (y/2) ( 1 + √2)  =  [ ( 1 + √2) / 2 ]   y

So...twice the distance is   (1 + √2) y  = distance between the parallel sides

Volume of a cone  =

(1/3) (3.14) (diameter /2)^2  (height)

(1/3) (3.14) (18/2)^2 (18)  =

(3.14/3) * (9)^2 (18)  =

3.14 *  81  * 18/3   =

3.14 * 81 * 6  ≈    1526.04 ft^3

Using your diagram, I can prove that  AX  = BY

Note that we have two triangles, ACX  and BCY

AC    =  CY   =  "b"

And

XC  = BC  =  "a"

And  angle ACX  = angle YCB

So.... by SAS   .... triangle  ACX  is congruent to triangle YCB

So... AX  =  BY

But...I have not figured out how to prove that CZ  is also equal to these

Maybe someone can provide the final step  ???

36a^4b^10 - 81a^16b^20   =

(6a^2b^5  + 9a^8b^10) ( 6a^2b^5 - 9a^8b^10)

We can find the common ratio, r,  by

Ratio  of   :  2nd term / 1st   term   =

6 / 9   =   2/3

So.....the explicit formula  is

a_n  = 9 (2/3)^{n- 1}

The period of the first graph is 5 seconds

The second graph  can be modeled by

y = 6sin (  2pi * x / 1.5    -  pi / 2)   +  6

Here's the graph : https://www.desmos.com/calculator/1hzzmtin1z

At..  x  = .75 s....we have

2pi * (.75) / 1.5   - pi.2   =

pi  - pi/2   =  pi/2    =  90°

a₁   = 10

a_n  =  -8 a_n-1

2.)  

a)  The profit of a company was $10 million in 2006 and $20.6 million in 2012. Find a linear model to represent the profit P of the company (in millions of dollars) in function of t . Variable t represents the number of year after 2004.

b)  Interpret the meaning of the model’s slope in the context of the problem. Include units in your interpretation. 

a)  We have the points   [ 2, 10]  and [ 8, 20.6]

The slope is    [ 20.6 - 10 ]  / [ 8 - 2 ]  =  10.6 / 6   =  53/30

The linear model is

y = (53/30) ( t - 2) + 10

y = (53/30)t  -  53/15  + 10

y =  (53/30)t + 97/ 15

b)  The slope, (53/30)  indicates the profit is increasing by  ≈  $ 1.76 M    per year

We need to find the x coordinate  of A, thusly

20 =  (-1/2)x^2 - x + 30

Subtract 20 from both sides  and rearrange

(-1/2)x^2 - x + 10  =  0       multiply through by -2

x^2  + 2x - 20  =  0

Using the quadratic formula to find the solutions  of this and taking the negative value, we have that

x  =  -1 - √21

So     ( -1 - √21, 20) is one point on the line

And to find the y values associated with B we have

(-1/2)(5)^2 - 5 + 30   =  -25/2 + 25   =   12.5

So   the point (5, 12.5)  is also on the graph

So  the slope is   [ 20 - 12.5 ]  / [  -1 - √21 - 5 ]  =  [ -7.5] / [ 6 + √21 ] 

So......the equation of the line  is

y  =  (  [ -7.5] / [ 6 + √21 ] )  ( x - 5)  + 12.5

OK.....but only if you can find the surface area of THIS, first......