Question

model 2 – comparing shapes

side	1 cm	2 cm	4 cm
volume	1 cm³	8 cm³	64 cm³
surface area - to - volume ratio			96:64 = 1.5:1

diameter	1 cm	2 cm	4 cm
volume	0.5 cm³	4.2 cm³	34 cm³
surface area - to - volume ratio

diameter × height	1 cm × 1 cm	1 cm × 2 cm	1 cm × 4 cm
volume	0.8 cm³	0.6 cm³	3.1 cm³
surface area - to - volume ratio

Explanation:

Step1: Calculate cube ratios

Cube (1 cm side):

Ratio = $\frac{\text{Surface Area}}{\text{Volume}} = \frac{6}{1} = 6:1$

Cube (2 cm side):

Ratio = $\frac{\text{Surface Area}}{\text{Volume}} = \frac{24}{8} = 3:1$

Step2: Calculate sphere ratios

Sphere (1 cm diameter):

Ratio = $\frac{\text{Surface Area}}{\text{Volume}} = \frac{3}{0.5} = 6:1$

Sphere (2 cm diameter):

Ratio = $\frac{\text{Surface Area}}{\text{Volume}} = \frac{13}{4.2} \approx 3.1:1$

Sphere (4 cm diameter):

Ratio = $\frac{\text{Surface Area}}{\text{Volume}} = \frac{50}{34} \approx 1.5:1$

Step3: Calculate cylinder ratios

Cylinder (1 cm × 1 cm):

Ratio = $\frac{\text{Surface Area}}{\text{Volume}} = \frac{4.7}{0.8} \approx 5.9:1$

Cylinder (1 cm × 2 cm):

Ratio = $\frac{\text{Surface Area}}{\text{Volume}} = \frac{7.9}{0.6} \approx 13.2:1$

Cylinder (1 cm × 4 cm):

Ratio = $\frac{\text{Surface Area}}{\text{Volume}} = \frac{14}{3.1} \approx 4.5:1$

Answer:

Completed Cube Table (Surface Area-to-Volume Ratio):

1 cm side: $6:1$
2 cm side: $3:1$
4 cm side: $1.5:1$

Completed Sphere Table (Surface Area-to-Volume Ratio):

1 cm diameter: $6:1$
2 cm diameter: $\approx 3.1:1$
4 cm diameter: $\approx 1.5:1$

Completed Cylinder Table (Surface Area-to-Volume Ratio):

1 cm × 1 cm: $\approx 5.9:1$
1 cm × 2 cm: $\approx 13.2:1$
1 cm × 4 cm: $\approx 4.5:1$