CPhill

Thanks, EP   !!!!!

Note  that if we  multiply  g(x)  by  -(1/2)  we get   -x^4 + 3x^2 -x + 1/2

Adding this  to  f(x) produces  -x + 5/2

So....the  smallest  possible  degree of  f(x) + b * g(x)   =    1

Area  =  W * L

160  =  W ( W + 6)      simplify

!60    =  W^2  + 6W     rearrange as

W^2  + 6W  - 160  =  0        factor

(W  + 16) ( W  - 10)  =  0  

Only  the  second factor produces  what we want   W -  10   =   0  ⇒   W   = 10  yards

a)   Since the population grew by 10.9% in the last 10 years......we have that

 A₀ * ( 1.109)  =  277    divide both sides  by  (1.109)

A₀  =   277 / (1.109)   ≈  250

b ) So   to find the annual growth  rate  we  can solve  for  this

277  = 250e ^(k * 10)      divide both sides  by  250

277/250   e ^(10k)       take the Ln of both sides

Ln  (277/250)  = Ln e^(10k)      and we can write

Ln (277/250)  = 10k * Ln e           Ln e   =1

LN (277/250)  / 10   =  k  ≈ .01025   =  annual  growth rate of ≈ 1.025 %

c)   400  = 277 e ^(.01025 * t)        divide both sides by 277

400 / 277  = e^(.01025 * t)       take the Ln again

Ln (400 /277)  = .01025 t     divide  both sides  by   01025

Ln (400/277) / .01025  = t ≈ 35.8   years from the first year =  25.8 years from now 

Good job, EP !!!!!

xy                             xz                          yz

_____    = 1        _______  =   2          _______   =   3

x + y                      x + z                        y + z

We can  convert these to this system:

x + y  =  xy      ⇒  xy - y  = x  ⇒   y ( x - 1)  = x   ⇒  y  = x  / (x - 1)         (1) 

2x + 2z = xz    ⇒   xz - 2z =  2x ⇒  z ( x - 2)  = 2x  ⇒  z   =  2x / (x - 2)     (2)

3y + 3z =  yz     (3)

Sub (1)  and (2)  into 3

3x / (x  -1)  + 3 (2x)  /(x - 2)   = (x * 2x) / [ (x - 1)(x - 2)]

Multiply through  by (x  -1) (x - 2)

3x ( x -2)  + 6x ( x -1)  =  2x^2        simplify

3x^2 - 6x  + 6x^2  - 6x = 2x^2

7x^2   - 12x  = 0      factor

x ( 7x  - 12)  = 0

The first factor   x  = 0  provides  no  solution

Setting the  second factor to 0  and solving for x produces  x = 12/7

And  y  = (12/7) / (12/7 -1)  =  (12/7) / (5/7)  =  12/5

And  z =   2 (12/7)/ (12/7 - 2)  =  (24/7) / (-2/7)  =  24/ -2   = -12

We can use  the Law  of Cosines here

31^2  = 39^2  +  22^2 - 2 (39 * 22) cos (ABC)

Rearrange as

cos (ABC) =   [ 31^2 - 39^2  - 22^2 ]  / [ -2 * 39 * 22 ]    =  87 / 143

Taking  the inverse cosine

arccos  ( 87 / 143 )  ≈  52.5°   =  x

Here's  the algorithm :

42 / 5  =   8  +  remainder

42 /  5^2  =  42 / 25  =   1 + remainder

Ignore the remainders.....add the integers.....

8 + 1   =  9

There  are  9  trailing zeroes  !!!!!

B. More individual voters cast their vote for Gore.

C. The contract provided changes in government that appealed to Americans at that time, leading to a majority of seats in Congress being held by Republicans.