CPhill

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Triangle on the left hypotenuse  =  sqrt [  5^2  + (15 - 3)^2  ] =  sqrt [ 5^2 + 12^2 ] =

sqrt [ 25 + 144 ]  = sqrt [ 169 ]  = 13 ft

Triangle on the right  hypenuse =  sqrt [ (14 - 5)^2  + (15 -3)^2 ]    sqrt [ 9^2 + 12^2 ]  =

sqrt [ 81 + 144]   =  sqrt [ 225]  = 15 ft

So...the rope length is   13 + 15  =   28 ft

Let W  be the number of workers [ per day ] that we need to make a profit

Total costs  =  500 +  15 * 8  * W      (1)

Total Revenue  =   5 ( 3.10) * 8 *  W    (2)

We want to solve this for W

(2) > (1)

5 ( 3.10) * 8 * W  > 500 + 15 * 8 * W     simplify

124W   > 500 + 120W   subtract  120 W from both sides

4W > 500     divide both sides by  4

W >  125

So...you are correct....we  need 126 workers  [ at a minimum ] to make a profit !!!

2^(3x - 2)  =  5^x      take the ln of both sides

ln  2^(3x - 2)  =  ln 5^x     and we can write

(3x - 2) ln (2)  =  x ln 5    simplify

3x ln 2  -  2ln 2   = x ln 5     rearrange

3x ln 2 - x ln 5   =  2ln 2       factor

x  [ 3 ln 2 - ln 5]  = 2ln 2      divide both sides by  [ 3 ln 2  - ln 5 ]

x  =       2 ln 2

         __________        correct  !!!!

          3ln 2 - ln 5

OK....glad to help....!!!!

Good question  !!!!....note that....

Every point on the graph should check...let's check  ( 4 , -2)

So we have

-2  =  log₂ (4 - 3)  - 2       add to to  both sides

0  =  log₂ (1)     ......  [ Note   log _a (1)  always  = 0   ]  [ proof...check this  log (1) / log (2) ]

0  =  0

True  !!!!

So...we know we have the  correct equation

y  is supplemental to the 131°  angle

So  y  =   180  - 131  = 49°

And this angle is vertical to the bottom left vertex angle of the top triangle  =  49

Tha angle supplemental to the 142° angle at the bottom is 180 - 142  = 38°

So angle  w   = 180  - 38  - 49  = 93°

And by the exterior angle theorem....w =  the sum of 42°  and the angle vertical to the bottom right vertex angle of the top triangle  

So this angle  =    93 - 42   =  51°

So....t  =  180  - 51  - 49  =  180 - 100   =  80°

See my more complete explanation  in the answer to your  latest  post

In   y  = log ( x - a)  + b

The ( x -a) part translates the graph left/right   and the "b" translates it up/down

(x - a)  translates the graph right  by a units

( x + a)  translates the graph left by  a units

+ b   translates the graph up by  b units

- b  translates the graph down by  b units

Notice that it appears the the  parent has been translated to the right by 3  units...

The thing that translates the parent to the right by 3  units must  be  " ( x - 3 ) "

So...it appears that the last answer is correct

To check this....note that  (7,0)  is on the graph...which means that

0  =  log₂ ( 7 - 3)  - 2    must be true....so....

0 =  log ₂ ( 4)  - 2        [  Note that  log₂ (4)  =  log 4 /  log 2   =  2  ]

0  =  2  -  2

Which is true  !!!!

Does this make sense  ???

OK.....I'll put them in  radical  form