Question

100 POINTS!!!

All the squares outlined on the model cells are 1 cm by 1 cm.



.Each face of Cube A measures 1 cm long by 1 cm wide.

.Each face of Cube B measures 2 cm long by 2 cm wide.

.Each face of Cube C measures 3 cm long by 3 cm wide.

.Calculate the area of one side of each cube by multiplying the length times the width.

.Calculate the surface area of each cube by multiplying the area of one side of the cube by 6 sides.

.Calculate the volume of each cube by multiplying length times width times height.

.Build the ratio for each cube: (surface area)/volume

.For each cube, divide surface area by volume and round off to the nearest whole number. For example, 1.5 to 1 is 3:2.

Answer 1

1) Area of one side of each cube

The area of one side of each cube can be calculated by multiplying the length times the width:

`A=L\cdot W`

where

L = length

W = width

For cube A:
`L=1 cm, W=1 cm`

So the area is:
`A_A = (1)(1)=1 cm^2`

For cube B:
`L=2 cm, W=2 cm`

So the area is:
`A_B=(2)(2)=4 cm^2`

For cube C:
`L=3 cm, W=3 cm`

So the area is:
`A_C=(3)(3)=9 cm^2`

2) Surface area of each cube

The surface area of a cube is given by the sum of the areas of all its faces. Since a cube has 6 identical faces, this means that the total surface area is equal to 6 times the area of one face:

`A=6 A_1`

where

`A_1` is the area of one face of the cube

For cube A, the area of one face is
`A_A=1 cm^2`

So the surface area of cube A is

`A'_A = 6A_A=6(1)=6 cm^2`

For cube B, the area of one face is
`A_B=4 cm^2`

So the surface area of cube B is

`A'_B = 6A_B=6(4)=24 cm^2`

For cube C, the area of one face is
`A_C=9 cm^2`

So the surface area of cube C is

`A'_C = 6A_C=6(9)=54 cm^2`

3) Volume of each cube

The volume of a cube is obtained by multiplying its length, its width and its height:

`V=L\cdot W \cdot H`

Where:

L = length

W = width

H = height

Moreover for a cube, all the sides have equal length, so
`L=W=H`

So the volume can be rewritten as

`V=L^3`

For cube A: L = 1 cm

So the volume is
`V_A=(1 cm)^3 = 1 cm^3`

For cube B: L = 2 cm

So the volume is
`V_B=(2 cm)^3 = 8 cm^3`

For cube C: L = 3 cm

So the volume is
`V_C = (3 cm)^3 = 27 cm^3`

4) Ratio surface area/volume for each cube

In this part, we have to calculate the ratio between surface area and volume of each cube:

`r=(A)/(V)`

where

A is the surface area

V is the volume

For cube A, we have:

`A_A = 6 cm ^2` (surface area)

`V_A=1 cm^3` (volume)

So the ratio for cube A is:

`r=(6)/(1)=6`

For cube B, we have:

`A_B = 24 cm ^2` (surface area)

`V_B=8 cm^3` (volume)

So the ratio for cube B is:

`r=(24)/(8)=3`

For cube C, we have:

`A_C = 54 cm ^2` (surface area)

`V_C=27 cm^3` (volume)

So the ratio for cube C is:

`r=(54)/(27)=2`