CPhill

The slant height of the cone  is  sqrt [ 3^2 + 4^2]  = sqrt [25]  =5

And its lateral surface area is given by pi * radius * slant height  =

pi * 4 * 5  =

20 pi   [units^2]   = A

The lateral surface are of the cylinder is given by

2pi * radius * height  =

2pi * 4 * 3  =

24pi  [units^2 ]  = B

So

A/B  =   20 pi / [24 pi]  =  20 / 24  =   5/6

[ x^2 + 4x + 3 ] / [ 2x - 1 ]

_____________________  =   [ invert the bottom fraction and multiply ]

[ x^2 + x ] / [ 2x^2 -3x + 1 ]

 [ (x + 1) (x + 3) / [2x -1 ]  *  [ 2x^2 - 3x + 1 ] / [x^2 + x]  =

 (x + 1) ( x+ 3) / ( 2x - 1)   *  [ (2x -1) (x -1)] / [ x ( x + 1) ]  =

(x + 1) ( x + 3) / [ x ( x + 1) ]  *  [ (2x - 1) (x - 1) ] / (2x -1)   =

(x + 3) / x   *   (x - 1)  =

(x + 3) (x -1)  /  x

Total costs  =  500 +  8 * 8x   = 

 Fixed cost + (hrs per day)( hourly wage rate) (x)  =

  500 + 64x      where x is the number of workers we need

Total Revenue  = 

(hrs per day) (number of widgets/hour)(sales price/widget) (x)  =

8 * 5 * 3.10 * x  =  124x

And we want to solve this

124x >  500 + 64x      subtract  64x from both sides

60x > 500

x > 500/60

x > 8.33  workers

So...we need 9 workers (at a minimum )

3x^2 + 7x  + c  = 0

If this has  2 real roots, the discriminant  [ 7^2 - 4(3)c ] is > 0

So

49 - 12c   >  0

49 > 12c

49/12  > c  means the same thing as

c < 49/12 

c < 4 + 1/12

And since  c must be a positive integer value then the possibilities for c  are given by

0 < c < 4 + 1/12

So....c  = 1, 2, 3, 4

And the product of these  =  24

Since  the numerator is a polynomial, it won't have any domain restrictions

The domain restrictions  will occur when an x value makes the denomiantor = 0

So...to find these values, set the denominator to 0  and solve for x

x^2 + 2x  - 3    = 0     factor

(x + 3) ( x - 1)  = 0

Set each factor to  0  and solve for x

x + 3  = 0                 x - 1  = 0   

x  = -3                       x  = 1

So  x cannot be  -3    or   1

Let

x1  =      [( 2m  + √ [ 4m^2  - 4(1)(m^2 + 2m + 3 ) ]  ) / 2 ]

x2  =  [( 2m  - √ [ 4m^2  - 4(1)(m^2 + 2m + 3 ) ]  ) / 2 ]

(x1)^2  + (x2)^2  =

2 * [4m^2 + 4m^2  - 4m^2 - 8m - 12 ] / 4  =

[4m^2 - 8m - 12 ] / 2  =

2m^2 - 4m - 6

The m value that minimizes this is   4 / (2 * 2)  =  4 / 4   = 1

So...the minimum value of  (x1)^2 + (x2)^2  = 2(1)^2 - 4(1) - 6  =  -8

We have these points on the graph  ( 4 , 1400)   and  (6 , 2100)

The slope between these points is  [ 2100 - 1400 ] / [ 6 - 4 ] = 700 / 2  = 350 / 1

This means that, for every additional student, the tuition cost is $350 more

We have these points on the graph  ( 0, -2)  and  ( 4, 0)

The slope is   [ 0 - -2 ] / [ 4 - 0 ]  =   2 / 4   =   1 / 2

Two points on the graph  are (0, -3)  and (-3, 0)

The slope  between these are  [ 0 - -3 ] / [ -3 - 0 ]  =  3 / -3   = -1

So....the equation of the line is

y   = -1(x - 0) - 3    

y =  -x - 3

If the polynomial  has one real root, the discriminant [ b^2 - 4ac ] must  = 0

So   a  =1, b  = c+ 1 ...so we have

(c + 1)^2  - 4(1)c    simplify

c^2 + 2c + 1  - 4c  = 0

c^2 - 2c + 1  = 0      factor the left side

(c - 1)^2  = 0         take the square root of both sides

c - 1  = 0

c  = 1