CPhill

3)  36x^3  + 96x^2  + 42x + 112

Take out the GCF  between  36x^3  and 96x^2  =  12x^2

Take out the GCF between 42x  and 112   = 14

So we have

12x^2 ( 3x + 8)  + 14 (3x + 8)  .....  (3x + 8)  is common...so....

(3x + 8) (12x^2 + 14)     ....note that we can factor a 2 out of the second factor...and we have...

(3x + 8)* 2* (6x^2 + 7)  =

2(3x + 8) (6x^2 + 7)

4)  42n^3 - 48n^2  - 336n + 384

Take out the GCF between 42n^3 and 48n^2  = 6n^2

Take out the GCF between 336n  and 384  = 48

So we have

6n^2 ( 7n - 8) - 48(7n -8)

(7n -8) (6n^2 - 48)     ...and we can factor a 6 out of the second factor

(7n - 8) * 6 * ( n^2 - 8)

6(7n - 8) (n^2 - 8)

(a)   We have two trapezoids at  each end of this figure.....their combined perimeters =  2 [ 2*9 + 7 + 13 ] = 76 mm

The figure in the middle is a sector of  a circle...its perimeter is :

2 pi * radius *  ( 44/360)  =

2 * pi * 9 * (44 / 360)  ≈  6.91  mm

So...the perimeter  is   [ 76 + 6.91 ] mm  =  82.91  mm

(b)  The two  circular areas have a  combined perimeter of  2 * pi * 4 * (240 / 360)  ≈ 16.76 cm

The area of the equilateral  triangle in  the  middle is     (1/2)* 4^2 * √3 / 2  ≈   6.93  cm 

So...the approximate perimeter is   [ 16.76 + 6.93 ] cm  =  23.69 cm 

square feet of pool  = 

pi *  (diameter / 2)^2  =

pi * ( 20 / 2)^2  =

pi * 10^2  =

100 pi  ft^2  ≈   314.16  ft^2

(c)

56  = 2 pi  * radius *  (145 / 360)

56 = 2 pi * radius*  (29/72)

56 =  pi * radius * (29/ 36)

56 = (29/36) pi  * radius      ..divide both sides by  (29/36) pi

56/  [ 29/36 * pi ]   =  radius  ≈ 22.13  in

(d)

Arc Length =  2 pi * 5  *  (150 /360)  =   10 (5/12) * pi    =  (5/6) * 5 pi  ≈  13.09 m

The total amount of money James has is   3a + 2b + 2  =    2a  + 3b  +  6

Set these equations equal

3a +  2b + 2  = 2a + 3b + 6    rearrange as

a - b  = 4

a = 4 + b

If  Judith arrives with the same amount of money as James....then twice their money = 6 items each

2 ( 3a + 2b + 2)   =  6a  + 6b     simplify

6a + 4b + 4   =  6a  + 6b     subtract  6a from both sides

4b + 4  = 6b        subtract   4b from both sides

4  = 2b       divide both sids by 2

2  = b ⇒   the pencils cost $2 

And the pens cost     a  = 4 + b  = 4  + 2  =  $6

So   (a,b)  = (6, 2)

Volume of the outer sphere =  (4/3)  pi  * radius^3  =  (4/3) pi * (2.8)^3

Volume of the inner sphere  = (4/3) pi * radius^3  = (4/3) pi * (2.1)^3

Volume between the spheres  = (4/3)pi  [ 2.8^3  - 2.1^3 ] ≈ 53.16 m^3

(a)  central angle  = 60....circumference  = 42 cm

Formula for arc length =    2 * pi * radius  * (central angle / 360 )

     42   =   2 * pi * radius * (60 / 360)

     42   =   2 * pi * radius (1/6)

     42  =  pi / 3 * radius      ......multiply both sides by  3 /pi

     42* 3 /pi   = radius  ≈  40.11 cm

b)  central angle  = 72........radius  = 6 ft

Arc length =   2 pi * 6 * ( 72 /  360)   =  12 pi *  (1/5)  = 12 / 5 * pi  ft  ≈  7.54 ft

a) Circumference  = 2pi * r  =  2 pi * (d / 2)   = pi * d  = pi * 18 in   ≈  56.55 in

b)  2 pi * (6)  =  12 pi in   ≈  37.70 in

Again....c  and d don't specify if these are radii or diameters....

a)      Perimeter   =  pi * r   =  pi * (d /2)    = pi  * (11/2)  =  11pi / 2 mm ≈ 17.28  mm

b)      Perimeter  =  pi * r   =  pi ( 3)   =  3 pi   ft  ≈  9.42 ft

On c and d   ...you don't specify if these are  diameters or radii

Mass =  Density  * Volume

         =  ( 1g / ml) * ( 10 ml)    =   10  g