CPhill

log₂ 256  - log₅ 625  = 

log 256 / log 2  - log 625/log 5  =

8  - 4    =

4

[y^2-16] / [y^2+16 ]

The only thing that would make this undefined is when the denominator   = 0 .....however......no real number  for  y makes this so....thus.......this is defined for all  real values of y

2x + 5y

 [(1,3), (1,8), (5,6), (10,3)]

(1,8) maximixes  the function  = 42

5N  - 3  =  3N - 5      simplify

2N  =   -2          divide by 2

N  = -1

sec ( -(168+ 2/15) )   ≈ -1.02184

D) zero......by the triangle inequality......the sum of any two sides must be greater than the remaining side to have a triangle.......but  3 + 1  = 4  ....so ......no triangle possible......

The shell method seems logical here.....we have

      4

2 pi ∫  (x + 1)[ 5 - (x + 4/x)]  dx  =

      1

      4

2 pi ∫   5x - x^2 - 4 + 5 - x - 4/x     dx

      1

       4

2 pi ∫   4x - x^2 + 1- 4/x     dx    =  [16137 / 1250]  pi units^3

      1

ln (x^2-1) - ln(x-1) = ln(6)      write as

ln [ (x^2 - 1) / (x - 1) ]  =  ln 6     we can forget the logs

(x^2 - 1) / (x - 1)  = 6     rearrange

x^2 - 1  =  6 (x - 1)  simplify

x^2 - 1  = 6x - 6

x^2 - 6x +  5  = 0     factor

(x - 1) ( x -5) = 0    set each factor to 0    and x  = 1  [reject because it makes one or more logs undefined]  or x  = 5  which is valid..

12 + 12 + 36 + 36 + 24  =  120   in

Volume = 5.9 * 3.7 * 4.5 = 98.235 in^3

Area =  2 [ 5.9 * 3.7  + 5.9 * 4.5 +  3.7 * 4.5 ]  =   130.06  in^2