Let's analyze the given problem.

We are dealing with a right triangular prism, and we need to find:

1. The length of a (the diagonal of the right triangle base of the prism) using the dimensions provided: $12 \, \text{in}$ and $9 \, \text{in}$.
2. The length of b (the diagonal of the prism extending between two non-adjacent corners), once we know a and another dimension, $7 \, \text{in}$.

Part (a): Finding a

Since a is the hypotenuse of the right triangle with legs $12 \, \text{in}$ and $9 \, \text{in}$, we can use the Pythagorean Theorem: $a^2 = 12^2 + 9^2$ Let me calculate that.

Part (b): Finding b

Once we have a, we know that b forms another right triangle where a and the side $7 \, \text{in}$ are the legs, and b is the hypotenuse: $b^2 = a^2 + 7^2$ I'll calculate both of these values.### Part (a): The length of a is $15.0 \, \text{inches}$.

Part (b):

The length of b (the length of the tape) is approximately $16.6 \, \text{inches}$, rounded to the nearest tenth.

Would you like further clarification on any step? Here are 5 related questions for deeper exploration:

1. How is the Pythagorean Theorem applied in 3D geometry problems like this one?
2. What would happen if one of the dimensions was different? How would it affect a and b?
3. Can you describe how the length of b changes as the height of the prism increases?
4. What is the significance of understanding diagonal lengths in real-world objects like boxes or prisms?
5. How can trigonometry extend these concepts to angles within the prism?

Tip: Always double-check units in geometric problems to ensure all dimensions are consistent.