CPhill

We can use the angle sum identitiy for the cosine to  check this  ........

cos  (θ  + π/2) = 

cos θ cos π/2  -  sin  θ sin π/2  =

cos θ *  ( 0 )  - sin θ  * (1)  =

-sin θ

So...this is an  identity  !!!

The midpoint of the line connecting these two points will lie on the line we seek...and this is

[ (1 + 9 ) / 2, ( 1 + 5) / 2 ]  =  [ 5, 3 ]

And the slope of this line connecting the given points is  [ 5 - 1 ] / [ [ 9 - 1]  = 4/8  =  1/2

And the line we seek is perpendicular to this line...so its slope is  - 2

And the equation of this line  is

y = -2 (x - 5) + 3

y = - 2x  + 10 + 3

y  = - 2x + 13

So...m  +  b  =   -2 + 13   =  11 

Here's a pic :  https://www.desmos.com/calculator/jdelhysrdy

f(x) = l 9 - x l    g(x)  = √x  - 1

a)  f ° g (1)   =     l  9  - (√1 - 1) l   =  l 9 - 0 l   = l 9 l  =   9

     f ° g(3)  =       l 9 - ( √3 - 1 l   =    l 10 - √3 l  ≈  8.268

     f ° g(5)  =       l 9 - ( √5 - 1 l   =   l 10  - √5  l  ≈  7.764

     f ° g(7)   =     l 9 - ( √7 - 1) l  = l 10 - √7 l  ≈  7.354

     f ° g(0)   =     l 9 - ( √0  - 1 ) l  =  l 10 l   =  10

b)  f ° f(2)  =   l  9  - l 9  - 2 l l  =  l 9 - l7l l  = l 9 - 7 l  =  2

     f ° f(5)  =   l 9  - l 9 - 5 l l  =   l 9 - l 4 l l  = l 9 - 4 l  =  5

     f ° f(-2)  =  l 9  - l 9 - - 2 l l  =  l 9 - l 11 l l = l 9 - 11l =   l - 2 l  =  2

c)  Definitely not.....consider  f ° f (10)  =    l 9 - l 9 -10l l  =  l 9 - l - 1 l l  =  l 9 -  1 l  = 8

But    l 10  l  =  10

a)  We can  apply a "3D" version of the Pythagorean Theorem here :

The longest stick that will fit is given by:

√[ 24^2  + 18^2 + 12^2  ]  = √1044  ≈ 32.31  in

b)  We needt to find the diagonal distance across the bottom of the box to first answer this....we can apply the Pythagorean Theorem  here :

This  is given by : √ [ 24^2 + 18^2  ] = √900  = 30

We can find the angle that the stick makes with the bottom of the box by using the inverse tangent

arctan  (  12 / 30)  =  angle  ≈  21.8°

Volume of prisim = 5 * 0.9 * 0.02  = 0.09  mi^3

Density =  Mass / Volume   =  894 / 0.09   ≈  9933.3   kg/mi^3

Population density  = Number. of  people  / Number of square miles  =

466488 / 97.9  ≈    4765 people  per square mile

0.12 (repetaing)  = 12/99 =   4/33

0.345 (repeating)  = 345/999  = 115/333

So 

4/33  + 115/333  =

[4* 333 + 115 * 33] / [ 33 *333] =    1709 / 3663  =   .466557   (repeating)

So...the sum is  33

Just like the other problem, take the  cube root of 4913   =  17 m

Here you go, NSS....

http://www.kentchemistry.com/links/Nuclear/ArtificalTrans.htm

V_cube = side^3

Therefore

729  = side^3         take the cube root of each side

³√ 729  =  side  =     9 [m]