When fully burned, natural gas produces approximately 1,000 BTU of heat per cubic foot of fuel, and propane produces about...

Phase 1: Owning the Dataset

Visual Representation

We'll use a comparison table to organize the fuel properties:

Key Relationships

• Usable BTU = Total BTU × Efficiency
• Natural gas: \(1,000 \times 0.90 = 900\) usable BTU per cubic foot
• Propane: \(2,500 \times 0.90 = 2,250\) usable BTU per cubic foot

Phase 2: Understanding the Question

We need to find costs where both fuels have the same cost per BTU of usable heat energy.

For equal cost per usable BTU:

• Natural gas: Cost per usable BTU = \(\mathrm{(Cost\ per\ cubic\ foot)} \div 900\)
• Propane: Cost per usable BTU = \(\mathrm{(Cost\ per\ cubic\ foot)} \div 2,250\)

Setting these equal:

\(\mathrm{(Natural\ gas\ cost)} \div 900 = \mathrm{(Propane\ cost)} \div 2,250\)

Cross-multiplying:

\(\mathrm{Natural\ gas\ cost} \times 2,250 = \mathrm{Propane\ cost} \times 900\)

\(\mathrm{Natural\ gas\ cost} \times 2.5 = \mathrm{Propane\ cost}\)

Key Insight: The propane cost per cubic foot must be exactly 2.5 times the natural gas cost per cubic foot.

Phase 3: Finding the Answer

Let's check each natural gas cost to see if 2.5 times that value appears in our choices:

If natural gas = $0.0035:

• Propane should be: \(\$0.0035 \times 2.5 = \$0.00875\)
• Is $0.00875 in our choices? No, continue.

If natural gas = $0.0070:

• Propane should be: \(\$0.0070 \times 2.5 = \$0.0175\)
• Is $0.0175 in our choices? Yes! ✓

? Stop here - we found our answer.

Phase 4: Solution

Verification: Let's confirm these values give equal cost per usable BTU:

• Natural gas: \(\$0.0070 \div 900 = \$0.00000778\) per usable BTU
• Propane: \(\$0.0175 \div 2,250 = \$0.00000778\) per usable BTU ✓

Final Answer:

• Natural gas cost per cubic foot: $0.0070
• Propane cost per cubic foot: $0.0175