Please help, and explain, so I can understand!!

83)

Volume of cube = S^3, where S = one side

V = 16^3 =4,096 mm^3 - volume of the cube

Volume of square pyramid =S^2 x h/3, where S=side, h=height

V =S^2 x h/3

4,096 =16^2 x h/3

4,096 = 256 x h/3 Divide both sides by 256

16 = h/3 Cross multiply

h = 48 mm - Height of the pyramid

48 / 16 = 3x - higher is the square pyramid than the cube.

81)

The answer is "B"

79)

Simplify the following:

-(3 x^6 - 3 x^5 + 2 x^2 - 5) + x^6 - 2 x^5 - x^3 - 7

-(3 x^6 - 3 x^5 + 2 x^2 - 5) = -3 x^6 + 3 x^5 - 2 x^2 + 5:

-3 x^6 + 3 x^5 - 2 x^2 + 5 + x^6 - 2 x^5 - x^3 - 7

Grouping like terms, x^6 - 3 x^6 + 3 x^5 - 2 x^5 - x^3 - 2 x^2 - 7 + 5 = (x^6 - 3 x^6) + (-2 x^5 + 3 x^5) - x^3 - 2 x^2 + (-7 + 5):

(x^6 - 3 x^6) + (-2 x^5 + 3 x^5) - x^3 - 2 x^2 + (-7 + 5)

x^6 - 3 x^6 = -2 x^6:

-2 x^6 + (-2 x^5 + 3 x^5) - x^3 - 2 x^2 + (-7 + 5)

3 x^5 - 2 x^5 = x^5:

-2 x^6 + x^5 - x^3 - 2 x^2 + (-7 + 5)

5 - 7 = -2:

-2 x^6 + x^5 - x^3 - 2 x^2 + -2

Factor -1 out of -2 x^6 + x^5 - x^3 - 2 x^2 - 2:

-(2x^6 - x^5 + x^3 + 2x^2 + 2)

78.  Volume of prism  = Area of Base * Height  

The base is a square, so its area is  6^2  = 36 in^2

The height  is 10 in

So the volume is   36 * 10  =  360 in^3

Volume of cylinder = pi * (diameter/ 2)^2 * height  = pi * (10/2)^2 * 6  = pi * 5^2 * 6  =

150pi in^3  ≈ 471.23 in ^3

So.....the cylinder has the greater volume

79

x^6 - 2x^5 - x^3 - 7  -  (3x^6 - 3x^5 + 2x^2 - 5)

Distribute  the  " - '   across the terms in the parentheses

x^6 - 2x^5 - x^3 - 7 - 3x^6 + 3x^5 - 2x^2 + 5       combine like terms

x^6 -3x^6 - 2x^5 + 3x^5 - x^3 - 2x^2 - 7 + 5

-2x^6 + x^5 - x^3 - 2x^2 - 2

80

The  oblique cylinder doesn't matter.....we can consider it to be just like a "normal" cylinder as far as solving for the radius...so we have...

Volume of cylinder  = pi * r^2 * height

36 pi  = pi * r^2 * 18       divide both sides by 18 pi

2 = r^2      take the square root of both sides

√2  = r  ≈ 1.4 in

I need help with all, but any help is great help!!! Thanks in advance! 

84

Volume of cylinder  =   pi *8^2 * 16  =  1024 pi  in^3     (1)

Volume of cone  = (1/3) pi * 4^2 * height  =  16/3 pi * height    (2)

If they have equal volumes.....set (1) and (2) equal  and solve for the cone's height

1024 pi    =  (16/3) pi * height        divide out pi

1024  = (16/3) * height       multiply both sides by 3/16

1024 (3/16)  = height 

192  in  = height

So.....the cone is    192/16  =  12 times as high as the cylinder

85

The cylinder (and hemisphere) both have a radius of 6....so....the total volume =

Volume of a cylinder  with a radius of 6 and height of 12   +  Volume of a hemisphere with a radius of 6

So  we have

pi  [  r^2 * height of cylinder   +    (1/2)(4/3) r^3  ]  =

pi  [ 6^2 * 12  +  (2/3)*6^3 ]  =

pi [ 432  + 144 ]  =

pi  [ 576 ]  ≈   1809.6 in ^3

This is what I did to get 85)

Volume of Hemisphere: \((2/3) \Pi r^3=(2/3)\Pi 3^3=56.5486677 \)

Volume of cylinder is \(\Pi r^2h=\Pi 3^2*12=339.29 \)

339.29+56.5486677=395.8 in.^3 (rounded to the nearest tenth) 

81. (5x)^1/2=square root of 5x, which is \(\sqrt5x, B\)