CPhill

Consider if we added the  line  y =  -1

Note that te graph  would cut this line in two places......so...

Two values of  c

T =   ( x^2  + (74 - x)^2 )^(1/2) / 2  +  ( x^2 + 63^2) / sqrt (2)

T  =   ( x^2  + x^2 - 148 + 5476 )^(1/2) / 2   +  ( x^2 + 3969)^(1/2) / sqrt (2)

T'     =   (1/4) ( 4x - 148)/ ( 2x^2 - 148x + 5476)^(1/2)  +  (2x) /[sqrt (2)(x^2 + 3969)^(1/2)]    = 0

       ( x - 37)                                   x

_____________________  +  ______________________     = 0  

(2x^2 - 148x + 5476)^(1/2)        sqrt (2) (x^2 + 3969)^(1/2)

Note  that  we  can write  the denomnator of the  first fraction as

2 ( x^2 - 74x + 2738)

So   we can write

(x - 37)                                                                    x

__________________________   +  _____________________   =   0

sqrt (2) ( x^2 - 74x + 2738)^(1/2)       sqrt (2) ( x^2 + 3969)^(1/2) 

(x - 37)                                                          -x

____________________    =              _________________           sqaure both sides

(x^2 - 74x + 2738)^(1/2)                        (x^2 + 3969)^(1/2) 

(x - 37)^2                                  x^2

_______________     =   ___________       cross-multiply

x^2 - 74x + 2738                x^2 + 3969 

(x - 37)^2 ( x^2 + 3969)  =   x^2 ( x^2 - 74x + 2738)     simplify

x^4 - 74 x^3 + 5338 x^2 - 293706 x + 5433561   =  x^4 - 74 x^3 + 2738 x^2

After a little manipulation we get

2600 x^2 - 293706 x + 5433561 = 0

Putting this into the quadratic formula  and  evaluating  we get  that  

x =   [ 293706   - sqrt  ( 293706^2  - 4*2600*5433561)]  / (5200)  =

[ 293706  -  sqrt ( 29754180036)  ]  / 5200  =

[ 293706  - 172494  ] /5200  =   23.31  / 100  =   23.31  = x

(There is another  value of x but it does not give us a minimum.......this is probably  due to the  fact that we squared the derivative )

Note that with weights of 1,2,4   we can weigh out any  amount from  1 - 7 inclusive

And

1, 2, 4 , 8   lets us weigh out any  amt  from1-15

And

1,2,4,8,16   lets us weigh out  any amt from 1 - 31

And

1,2,4,8,16,31  lets us weight out any  amt  from 1 - 63

Note  that  8 pounds = 8 * 16  ounces  =  128  ounces

1,2,4,8,16,32,64   allows us to weigh out any amt  from 1 - 127  ounces

You have the correct Total Time function

We could use Calculus  to solve this but the  derivative is a little messy....I solved it  here  with a graph :

https://web2.0calc.com/questions/interesting-geometry

However...if you want me to, I  think I can solve it with pure Calculus.....

I won't perform the  computations....but here is a way to tell

put (-6)  into the function  and evaluate

put (-5)  into the function and evaluate.....if the signs on the reults are different, then a zero occurs between these values

If you  have to  ...do  the  same  with   -5   and  -4 ,   -4 and - 3  and    -3, and  -2

If you  do this correctly, you should find that   -4  and  -3   is correct  !!!

False......the 2x^3  term determines the end behavior  on the left....as x becomes moroe and more negative.....the function becomes more and more negative  towards - inf

False....the 2 x^4 term  determines the end behavior on the  right of the  graph.....as x  becomes more and more positive,  becomes more and  more positive towards + inf

False  -9x^5   becomes more  positive as  x is more and more negative.....the limit  towards -inf  =  + inf

We have the form

y= asin (x + b)

The function is

y = -4sin( x+ pi/2)

The amplitude = l a l  =  l -4 l   = 4

Also.....The midline  is y = 0

The amplitude is the height of the curve  above the midline =  4

See the graph here :

https://www.desmos.com/calculator/o06yqc9wkk

4x + y  - 24   =  0

x intercept  =  6           y intercept  =  24

So   the points  are   ( 6,0)  and ( 0,24)

The midpoint of this segment =    ( 3, 12)

And the  slope of this segment =  -24/6 =  -4

So.....the equation of the  perpendicular  bisector is

y= (1/4) (x - 3) + 12

4y  = x - 3 +  48

4y  =  x  + 45

x - 4y + 45  =  0

When x = 11  then  

11  - 4a  + 45  =  0

56- 4a  =  0

56  = 4a

a  = 56/4  =  14

See the graph here :  https://www.desmos.com/calculator/bi4jwkbp07

Use the distance formula (squared)

(2 - -1)^2  + ( 1 - n)^2   =   25          simplify

9  + n^2  - 2n + 1   =  25

n^2  - 2n + 10  =  25

n^2 - 2n  -  15  =  0        factor 

(n - 5) ( n + 3)  =  0

Setting each factor to 0and solving for  n produces   n = 5   and  n =  -3

Their product =  -15

First one   :   -5pi/6 =  -150    ....360 - 150 = 210 degrees.....  Correct

Second one  :   45pi/7  =   6 pi  + 3pi/7   :   Correct

Third  :  -274 degrees  =  360  - 274  = 86 degrees  ......Correct  !!!

Last    478 - 360 =   118 degrees  = 2nd Q   ....Correct  !!!!

Good job   !!!!