Let's solve the problem step by step.

Part (a): Find $a$

The given figure represents a rectangular prism with dimensions 5 meters, 12 meters, and 10 meters. To find $a$, we need to use the Pythagorean theorem to calculate the diagonal of the base (a right triangle formed by the sides 5 meters and 12 meters).

The formula for the diagonal $a$ is:

$a = \sqrt{(5^2 + 12^2)}$

$a = \sqrt{(25 + 144)} = \sqrt{169} = 13 \, \text{meters}$

Thus, $a = 13 \, \text{m}$.

Part (b): Find $b$

Now, we use the diagonal $a$ (which is 13 meters) and the height of the prism (10 meters) to find the length of the branch $b$, which is the diagonal of the 3D shape. This also forms a right triangle, where $a$ and the height are the legs, and $b$ is the hypotenuse.

The formula for $b$ is:

$b = \sqrt{(a^2 + 10^2)}$

Substitute $a = 13 \, \text{m}$:

$b = \sqrt{(13^2 + 10^2)} = \sqrt{(169 + 100)} = \sqrt{269}$

$b \approx 16.4 \, \text{meters}$

Thus, $b = 16.4 \, \text{m}$ (rounded to the nearest tenth).

Do you need further details or have any questions?

Here are 5 related questions you could explore:

1. What would the length of the diagonal be if the height of the prism was changed?
2. How does the Pythagorean theorem extend to three dimensions?
3. How can you find the surface area of the rectangular prism in this example?
4. What would the branch length $b$ be if the base dimensions were 8 m and 6 m?
5. How can you use vector geometry to calculate the diagonal of a rectangular prism?

Tip:

When solving for diagonals in three-dimensional shapes, always visualize the triangle formed by the edges to apply the Pythagorean theorem correctly.