Terry needs to purchase some pipe for a plumbing job that requires pipes with lengths of 1 ft 4 in,...

1. Translate the problem requirements: Convert all pipe lengths from feet and inches to a single unit (inches) for easier comparison and combination. Identify that we need to find the minimum number of 10-ft (120-inch) pipes to accommodate all required lengths.
2. Calculate total length needed: Add up all required pipe lengths to establish a baseline minimum and check if a simple division approach works.
3. Apply bin packing strategy: Since we can't simply divide total length by pipe length (due to cutting constraints), group the required lengths to fit optimally within 120-inch segments, starting with the longest pieces first.
4. Verify the minimum configuration: Confirm that our grouping uses the fewest possible 10-ft pipes and that no better combination exists.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're working with. Terry needs specific pipe lengths, and we need to figure out how many 10-foot pipes to buy so he can cut them to get all the pieces he needs.

First, let's convert everything to inches since mixing feet and inches gets confusing. Remember: 1 foot = 12 inches.

Required pipe lengths:

• 1 ft 4 in = \((1 \times 12) + 4 = 16\) inches
• 2 ft 8 in = \((2 \times 12) + 8 = 32\) inches
• 3 ft 4 in = \((3 \times 12) + 4 = 40\) inches
• 3 ft 8 in = \((3 \times 12) + 8 = 44\) inches
• 4 ft 8 in = \((4 \times 12) + 8 = 56\) inches
• 5 ft 8 in = \((5 \times 12) + 8 = 68\) inches
• 9 ft 4 in = \((9 \times 12) + 4 = 112\) inches

Each 10-foot pipe from the store = \(10 \times 12 = 120\) inches

Process Skill: TRANSLATE - Converting mixed units to a single unit makes calculations much clearer

2. Calculate total length needed

Let's add up all the required lengths to see how much pipe Terry needs in total:

\(16 + 32 + 40 + 44 + 56 + 68 + 112 = 368\) inches

Now, if we could just buy pipe by total length, we'd need: \(368 \div 120 = 3.07\) pipes

Since we can't buy a fraction of a pipe, we'd need at least 4 pipes. But wait - this assumes we can perfectly use every inch of pipe with no waste. In reality, we might not be able to fit the pieces together perfectly within each 120-inch section.

For example, the 112-inch piece is very long and leaves only 8 inches remaining in a 120-inch pipe. We need to see if our other required pieces can actually fit in those leftover spaces.

3. Apply bin packing strategy

This is like trying to pack different sized boxes into shipping containers - we want to minimize waste. Let's start with the longest pieces first and see how we can group them.

Pipe #1: Start with the longest piece

• 112 inches (leaves \(120 - 112 = 8\) inches remaining)
• Can we fit anything else? Our smallest piece is 16 inches, which won't fit in 8 inches
• Pipe #1 total: 112 inches

Pipe #2: Next longest pieces

• 68 inches (leaves \(120 - 68 = 52\) inches remaining)
• 44 inches fits! (leaves \(52 - 44 = 8\) inches remaining)
• Nothing else fits in 8 inches
• Pipe #2 total: \(68 + 44 = 112\) inches

Pipe #3: Continue with remaining pieces

• 56 inches (leaves \(120 - 56 = 64\) inches remaining)
• 40 inches fits! (leaves \(64 - 40 = 24\) inches remaining)
• 16 inches fits! (leaves \(24 - 16 = 8\) inches remaining)
• Pipe #3 total: \(56 + 40 + 16 = 112\) inches

Pipe #4: Last remaining piece

• 32 inches (leaves \(120 - 32 = 88\) inches remaining)
• Pipe #4 total: 32 inches

All pieces accounted for: ✓ 112, ✓ 68, ✓ 44, ✓ 56, ✓ 40, ✓ 16, ✓ 32

Process Skill: VISUALIZE - Thinking of this as a packing problem helps us see why we can't just divide total length by pipe length

4. Verify the minimum configuration

Let's double-check that we truly need 4 pipes and can't do it with just 3.

We have 3 pipes of 120 inches each = \(3 \times 120 = 360\) inches total
We need 368 inches total

Even though \(360 < 368\), let's see if we can rearrange the pieces differently:

The 112-inch piece must go in its own pipe (since \(112 + 32 = 144 > 120\)).
That leaves us 2 pipes (240 inches) for the remaining pieces totaling \(368 - 112 = 256\) inches.
Since \(256 > 240\), it's impossible to fit everything in just 3 pipes.

Therefore, 4 pipes is indeed the minimum.

Process Skill: CONSIDER ALL CASES - We verified that 3 pipes is impossible before concluding 4 is the minimum

4. Final Answer

Terry needs to purchase a minimum of 4 ten-foot pipe lengths.

The answer is B. 4

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the constraint structure: Students often think this is simply a division problem - total length needed divided by pipe length available. They miss that this is actually a "bin packing" optimization problem where the specific sizes of individual pieces matter, not just the total length.

2. Unit conversion oversight: Students may attempt to work with mixed units (feet and inches) throughout the problem, leading to calculation errors and confusion. They might also forget to convert the 10-foot store pipes to inches when comparing with the required lengths.

3. Ignoring waste constraints: Students may not realize that leftover pieces from cutting cannot always accommodate the remaining required lengths, assuming that any remainder can be perfectly utilized.

Errors while executing the approach

1. Arithmetic errors in unit conversion: Students frequently make mistakes when converting mixed units like "3 ft 8 in" to inches, either forgetting to multiply feet by 12 or adding incorrectly (e.g., calculating 3 ft 8 in as \(3 \times 12 + 8 = 44\), but writing down 40).

2. Poor bin packing execution: Students may not systematically try different combinations of pieces within each pipe, or they might miss optimal groupings by not starting with the largest pieces first. This leads to suboptimal arrangements requiring more pipes than necessary.

3. Incomplete verification: Students may find one valid arrangement using 4 pipes but fail to verify whether 3 pipes could work, or conversely, they may conclude 3 pipes is impossible without properly testing all reasonable combinations.

Errors while selecting the answer

1. Selecting the theoretical minimum: Students who calculate \(368 \div 120 = 3.07\) may round up to 4 but then mistakenly select 3, thinking that since the decimal is small (0.07), the theoretical minimum of 3 pipes should work in practice.

2. Confusion between different calculation methods: Students might calculate both the total-length approach (giving 3.07 pipes) and the bin-packing approach (giving 4 pipes) but then select the wrong answer by choosing the result from the simpler but incorrect total-length method.