find the ratio of the volume to the surface area of the rectangular prism shown.

L= 4x+8
W= x+2
H= 2x+4

please show all your steps please!!

Just plug in the value:

(4x+8)(x+2)(2x+4) / 2((4x+8)(x+2)+(4x+8)(2x+4)+(x+2)(2x+4))

= 8(x+2)^3 / 28(x+2)^2

= 2(x+2)/7

To find the ratio of the volume to the surface area of the rectangular prism, you first need to calculate the volume and the surface area separately, and then divide the volume by the surface area.

Step 1: Calculate the Volume
The volume (V) of a rectangular prism is given by the formula V = L × W × H, where L is the length, W is the width, and H is the height.

Given:
L = 4x + 8
W = x + 2
H = 2x + 4

Substitute the given values into the volume formula:
V = (4x + 8) × (x + 2) × (2x + 4)

Simplify the equation:
V = (4x + 8)(x + 2)(2x + 4)
= (4x^2 + 16x + 32)(2x + 4)
= 8x^3 + 48x^2 + 96x + 64x + 128
= 8x^3 + 48x^2 + 160x + 128

So, the volume of the rectangular prism is 8x^3 + 48x^2 + 160x + 128.

Step 2: Calculate the Surface Area
The surface area (SA) of a rectangular prism is given by the formula SA = 2(LW + LH + WH), where L is the length, W is the width, and H is the height.

Given:
L = 4x + 8
W = x + 2
H = 2x + 4

Substitute the given values into the surface area formula:
SA = 2((4x + 8)(x + 2) + (4x + 8)(2x + 4) + (x + 2)(2x + 4))

Simplify the equation:
SA = 2(4x^2 + 8x + 8x + 16 + 8x + 16 + 4x^2 + 8x + 2x + 16)
= 2(8x^2 + 24x + 48)
= 16x^2 + 48x + 96

So, the surface area of the rectangular prism is 16x^2 + 48x + 96.

Step 3: Calculate the Ratio
To find the ratio of the volume to the surface area, divide the volume by the surface area:

Ratio = V / SA
= (8x^3 + 48x^2 + 160x + 128) / (16x^2 + 48x + 96)

This expression represents the ratio of the volume to the surface area of the rectangular prism.