Since shapes A and B are similar, the relationship between their volumes and surface areas follows certain proportionalities based on their linear scale factor.

Step 1: Relationship between volumes

The ratio of the volumes of two similar shapes is the cube of the ratio of their corresponding linear dimensions. The given volume ratio is:

$\frac{\text{Volume of shape A}}{\text{Volume of shape B}} = \frac{1}{10}$

Thus, the linear scale factor between the shapes is the cube root of the volume ratio:

[ \text{Linear scale factor} = \sqrt[3]{\frac{1}{10}} = \frac{1}{\sqrt[3]{10}} ]

Step 2: Relationship between surface areas

The surface areas of similar shapes are proportional to the square of the linear scale factor. So, the ratio of the surface areas is:

[ \frac{\text{Surface area of shape A}}{\text{Surface area of shape B}} = \left(\frac{1}{\sqrt[3]{10}}\right)^2 = \frac{1}{\sqrt[3]{100}} ]

Step 3: Calculate the surface area of shape A

We know the surface area of shape B is 72 cm², so the surface area of shape A can be calculated as:

$\text{Surface area of shape A} = 72 \times \frac{1}{\sqrt[3]{100}}$

Now, let's compute this expression.

$\sqrt[3]{100} \approx 4.6416$

$\text{Surface area of shape A} = \frac{72}{4.6416^2} \approx 72 \times 0.04641 \approx 3.345$

Thus, the surface area of shape A is approximately:

$\boxed{3.35 \, \text{cm}^2}$

Do you have any questions, or would you like further details?

Here are 5 related questions:

1. How do we derive the linear scale factor from the volume ratio?
2. Can you calculate the surface area ratio if the volume ratio were 1:27?
3. What happens to the surface area if the volume of shape A is doubled?
4. How does the surface area relate to the volume in similar 3D shapes?
5. Can you find the linear scale factor if the surface area ratio is given instead of the volume ratio?

Tip: Always remember that the ratio of surface areas of similar shapes is the square of the ratio of their linear dimensions!