To solve this problem of finding the distance from one corner to the opposite corner in a rectangular prism, we can use the 3D version of the Pythagorean Theorem.

The distance $d$ between two opposite corners of a rectangular prism is given by the formula:

$d = \sqrt{l^2 + w^2 + h^2}$

where:

• $l$ is the length,
• $w$ is the width,
• $h$ is the height.

For Part (a):

• Length $l = 11 \, \text{mm}$
• Width $w = 5 \, \text{mm}$
• Height $h = 4 \, \text{mm}$

We plug these values into the formula:

$d_a = \sqrt{11^2 + 5^2 + 4^2} = \sqrt{121 + 25 + 16} = \sqrt{162}$

$d_a = 12.73 \, \text{mm}$

For Part (b):

• Length $l = 8 \, \text{m}$
• Width $w = 0.2 \, \text{m}$
• Height $h = 0.1 \, \text{m}$

Using the same formula:

$d_b = \sqrt{8^2 + 0.2^2 + 0.1^2} = \sqrt{64 + 0.04 + 0.01} = \sqrt{64.05}$

$d_b = 8.004 \, \text{m}$

Summary:

• Distance for part (a) $d_a \approx 12.73 \, \text{mm}$
• Distance for part (b) $d_b \approx 8.004 \, \text{m}$

Would you like further details or have any questions?

Here are 5 related questions:

1. How do you calculate the diagonal of a 2D rectangle?
2. What would happen if one of the dimensions of the prism were zero?
3. Can you apply the Pythagorean Theorem to irregular shapes?
4. How does the Pythagorean Theorem extend to n-dimensional space?
5. Why is the diagonal always the longest side in a right-angled shape?

Tip: Always ensure units are consistent when solving problems, especially when mixing measurements like mm and m!