CPhill

2.After receiving his business degree from UNC-Chapel
Hill, John is offered positions with two companies. Company
A offers him $80,000 per year, with a $1,000 increase every
year. Company B offers him $60,000 per year with a 4%
increase every year. After how many years will the Company
B salary be higher than Company A?

We can solve this  :

80000 + 1000x  =  60000(1.04)^x

A graph seems  the  best way to go here

See here : https://www.desmos.com/calculator/6uzj1wumyw

The graph shows  that both companies will  pay the same at about 10.4 years

So...Company B will pay more after about 11 years

First one

We either  have  that

x + 5 ≤ 6.5                                                     or             x + 5 ≥ -6.5

Subtract 5 from both sides                                          Subtract 5 from both sides

x ≤ 1.5                                                                            x ≥  -11.5

So....the  solution  is  [ -11.5  ≤ x  ≤ 1.5 ]

\(2\log{(a -2b)} = \log{a} + \log{b}\)

We  can write 

log ( a - 2b)^2  =  log (ab)      

We have the  logs the same......we can solve this

(a - 2b)^2  =  ab

a^2 - 4ab + 4b^2  = ab

a^2  - 5ab + 4b^2  = 0       factor  as

(a - 4b) ( a - b)  =  0

So  either

a - 4b  = 0       or         a - b  =  0

a = 4b                         a = b

a/b   = 4                      a/b  = 1

THX, geno......here's another way

The sum of  the roots  =  -(-5)/1  =  5  =  a + b

The product of the  roots =  143/1 =  143  = ab

So

(a - 1) (b - 1)   =

ab  - a - b +  1  =

ab - (a + b)  + 1  =

143  - (5)  +  1  =

139

a)  (-4, 4)

r =  sqrt [ (-4)^2 + 4^2]  =  sqrt [32]  =  4sqrt (2)

θ  =  arctan (4/-4)  = arctan (-1)  =  -pi/ 4     ....this is a point in te second quadrant  so θ = -pi/4 + pi =

3pi/4

So   (r, θ)  =  ( 4sqrt 2, 3pi/4)

(b)  x^2 + (y - 5)^2   = 25      expand

x^2  + y^2 - 10y + 25 =  25      subtract 25 from both sides

x^2  + y^2  - 10y   =  0

r^2cos^2θ + r^2sin^2θ - 10 ( r sinθ)  = 0

r^2 ( cos^2θ + sin^2θ)  =  10r (sin θ)

r^2 (1)  = 10r (sin θ)        divide out r

r  = 10sinθ

Excellent, geno  !!!!

x - y  =  23

Square both sides

x^2 - 2xy + y^2  = 529      (1)

xy   = 23

2xy  = 46      (2)

Sub (2)  into (1)

x^2  -2(46) + y^2  = 529

x^2  - 92  + y^2   =  529       add 92 to both sides

x^2  + y^2  =  621

31° south of west =  180 + 31   =  211°

The (approx) coorrdinates  = 

[ (3)(57) cos (211°),  (3)(57) sin (211°) ]  =

[ 171 cos (211°) , 171 sin (211°) ] =

(-146.58, -88.07 )

We can  only  have  a repeated  root  whenever  the discriminant  =  0

First one   18^2 - 4(-1)(81)  > 0    so no repeated root

Second one     (-3)^2  - 4(3) (-168)  >  0    so no repeated root

Third one   (-4) - (4) (1) (-4)  > 0      so no repeated root

Fourth one  (-30)^2  - 4(25) (9)   =  900 - 100(9)  =  0      repeated root

Note  this factors  as  (5x - 3)^2

(5x - 3)^2  = 0     take both roots

5x - 3  = 0

5x  = 3

x  =  3/5   = repeated root

We  have  that  135°  =  (3/4)pi  rads

The area  of the  sector has the form

A =  (1/2) r^2 ( theta in rads)

So

54 pi  =  (1/2) r^2 [ 3/4] pi       divide out pi

54  = (1/2) (3/4) r^2

54  = (3/8) r^2        divide both sides by 3

18  = (1/8) r^2     multiply both sides by 8

144 = r^2       take the positive  root

12 (in)  = r