CPhill

First one

x^2 * log_x 27 * log₉ x  = x + 4      we can write

x^2  (log 27 / log x ) ( log x / log 9)  =  x + 4

x^2 (log 27 / log 9)  =  x + 4

x^2 ( log 3^3 / log3^2)  =  x + 4

x^2  *(3/2)  = x + 4

(3/2)x^2 - x - 4  = 0

3x^2 - 2x - 8   =  0     factor

(3x + 4) ( x - 2)  =  0

Setting each factor to 0   and solve for x  and we have that x = -4/3  or x  = 2

Only the second solution is good.....the first would mean that we were taking the log of a negative number in the original equation

Area of sector - Area of triangle  = Desired Area

The "formula" for the area is :

 r^2  [ (pi * 125/360)  -  (1/2) sin (125°)  ]  =

8^2 [  (pi * 125/360)  - (1/2)sin (125° ) ]  ≈  43.6  units^2

Thaks, helperid...here's an alternative solution

Without a loss of generailty we can let  AB  = 1

So   AHI  =   (1/2)(4)*2 sin (60)  =  4√3       (1)

Area  of  AFG  =  (1/2)(3)^2 sin (60) = (9/4)√3    (2)

So area of trapezoid  FGIH  =   √3 [ 4 - 9/4]  = √ 3 [ 7/4]

So  ratio of area of trapezoid FGIH  to AHI  =

√3 {7 /4]  / [  4√3 ]   =   7 / 16

EDIT TO CORRECT AN IDIOTIC ERROR  !!!!

Positive  on (-4, 0) and  (6,8) 

Increasing on  (-inf, 0)  and (1,2)

Constant on (-4, 0).....the function does not rise or fall in this interval

a)  The function we need is this :

P(t)  =  5655 (1.014)^t

b) In 2022, t  = 12......so we have

P(12)  = 5655 (1.014)¹²      ≈  6682  

The normal  graph is shifted to the left by 4/3 units  and is up by 3

So....the correct graph is  the one on the bottom left

Just go to the calculator here  and hover your cursor over the "log"  button

The "Ln"  function wil appear

Note...."log"  uses base 10

"Ln"  uses the constant "e" ≈  2.718.......