To determine how many feet of piping is required for all three sections, we need to simplify each term and then sum them up.
Let's start by simplifying each pipe length:
1. The first section of pipe is [tex]\( 6 \sqrt{96} \)[/tex] feet long.
- Simplify the term inside the square root: [tex]\( 96 = 16 \times 6 \)[/tex].
- Rewrite [tex]\( \sqrt{96} \)[/tex] as [tex]\( \sqrt{16 \times 6} = \sqrt{16} \times \sqrt{6} = 4 \sqrt{6} \)[/tex].
- Therefore, [tex]\( 6 \sqrt{96} = 6 \times 4 \sqrt{6} = 24 \sqrt{6} \)[/tex].
2. The second section of pipe is [tex]\( 12 \sqrt{150} \)[/tex] feet long.
- Simplify the term inside the square root: [tex]\( 150 = 25 \times 6 \)[/tex].
- Rewrite [tex]\( \sqrt{150} \)[/tex] as [tex]\( \sqrt{25 \times 6} = \sqrt{25} \times \sqrt{6} = 5 \sqrt{6} \)[/tex].
- Therefore, [tex]\( 12 \sqrt{150} = 12 \times 5 \sqrt{6} = 60 \sqrt{6} \)[/tex].
3. The third section of pipe is [tex]\( 2 \sqrt{294} \)[/tex] feet long.
- Simplify the term inside the square root: [tex]\( 294 = 49 \times 6 \)[/tex].
- Rewrite [tex]\( \sqrt{294} \)[/tex] as [tex]\( \sqrt{49 \times 6} = \sqrt{49} \times \sqrt{6} = 7 \sqrt{6} \)[/tex].
- Therefore, [tex]\( 2 \sqrt{294} = 2 \times 7 \sqrt{6} = 14 \sqrt{6} \)[/tex].
Now, add the simplified lengths of all three sections:
[tex]\[
24 \sqrt{6} + 60 \sqrt{6} + 14 \sqrt{6} = (24 + 60 + 14) \sqrt{6} = 98 \sqrt{6}
\][/tex]
Therefore, the total length of piping required is [tex]\( 98 \sqrt{6} \)[/tex] feet.
So, the correct answer is:
B. [tex]\( 98 \sqrt{6} \)[/tex] feet.
