To solve the problem:

Part (a)

We need to find the diagonal a of the rectangular base of the prism. The base dimensions are given as 5 m and 12 m. Using the Pythagorean theorem for the base rectangle: $a = \sqrt{(5^2 + 12^2)}$

Part (b)

To find the diagonal b, which extends from one corner of the prism to the opposite corner, we use the 3D version of the Pythagorean theorem: $b = \sqrt{(a^2 + h^2)}$ where $h = 15$ m is the height of the prism.

Step-by-step Solution:

1. Find $a$: $a = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \, \text{m.}$

2. Find $b$: $b = \sqrt{a^2 + h^2} = \sqrt{13^2 + 15^2} = \sqrt{169 + 225} = \sqrt{394}.$ Approximating to the nearest tenth: $b \approx 19.8 \, \text{m.}$

Final Answers:

• (a) $a = 13 \, \text{m}$
• (b) $b \approx 19.8 \, \text{m}$

Would you like further details or have additional questions?

Related Questions:

1. How does the Pythagorean theorem extend to 3D space?
2. What is the significance of rounding results in practical applications like construction?
3. Could we compute $b$ directly without finding $a$ first?
4. How do you determine when to use approximations versus exact values?
5. What are some real-life examples where calculating 3D diagonals is essential?

Tip:

When solving problems involving diagonals in three dimensions, always break them into smaller right triangles for clarity and accuracy.