When fully burned, natural gas produces approximately 1,000 BTU of heat per cubic foot of fuel, and propane produces about...
GMAT Two Part Analysis : (TPA) Questions
When fully burned, natural gas produces approximately 1,000 BTU of heat per cubic foot of fuel, and propane produces about 2,500 BTU of heat per cubic foot of fuel. For a furnace in which either natural gas or propane may be burned as fuel, the efficiency of a given fuel is the usable heat energy produced when the fuel is burned in the furnace, expressed as a percentage of the total heat energy produced when the fuel is burned. Kaiser will purchase a furnace whose efficiency with respect to either natural gas burned alone or propane burned alone is 90%.
In the table select for natural gas cost per cubic foot and for propane cost per cubic foot the values that are jointly consistent with the information given for which the fuel cost per BTU of usable heat energy produced by this furnace would be approximately the same for each fuel burned alone. Make only two selections, one in each column.
Phase 1: Owning the Dataset
Visualization
Let's create a comparison table to organize our fuel information:
| Property | Natural Gas | Propane |
| Heat per ft³ | 1,000 BTU | 2,500 BTU |
| Efficiency | 90% | 90% |
| Usable heat per ft³ | ? | ? |
| Cost per ft³ | ? | ? |
| Cost per BTU | Must be equal | Must be equal |
Calculate Usable Heat
With 90% efficiency:
- Natural gas usable heat: \(1,000 \times 0.90 = 900\) BTU per cubic foot
- Propane usable heat: \(2,500 \times 0.90 = 2,250\) BTU per cubic foot
Updating our table:
| Property | Natural Gas | Propane |
| Heat per ft³ | 1,000 BTU | 2,500 BTU |
| Efficiency | 90% | 90% |
| Usable heat per ft³ | 900 BTU | 2,250 BTU |
| Cost per ft³ | ? | ? |
| Cost per BTU | Must be equal | Must be equal |
Phase 2: Understanding the Question
We need cost per BTU of usable heat to be approximately the same for both fuels.
For any fuel:
\(\mathrm{Cost\,per\,BTU} = \frac{\mathrm{Cost\,per\,cubic\,foot}}{\mathrm{Usable\,BTU\,per\,cubic\,foot}}\)
Setting these equal:
- Natural gas: Cost per ft³ ÷ 900 = Cost per BTU
- Propane: Cost per ft³ ÷ 2,250 = Cost per BTU
Therefore: \(\frac{\mathrm{Natural\,gas\,cost}}{900} = \frac{\mathrm{Propane\,cost}}{2,250}\)
Rearranging: \(\mathrm{Natural\,gas\,cost} = \mathrm{Propane\,cost} \times \left(\frac{900}{2,250}\right)\)
Calculating the ratio: \(900 ÷ 2,250 = 0.4\)
Key insight: \(\mathrm{Natural\,gas\,cost} = 0.4 \times \mathrm{Propane\,cost}\)
Phase 3: Finding the Answer
We need to find which values from the choices satisfy: \(\mathrm{Natural\,gas\,cost} = 0.4 \times \mathrm{Propane\,cost}\)
Checking systematically:
If propane = $0.0175:
- Natural gas should be: \(\$0.0175 \times 0.4 = \$0.0070\)
- Is $0.0070 in our choices? Yes! ✓
Let's verify the cost per BTU is equal:
- Natural gas: \(\$0.0070 ÷ 900 = \$0.00000778\) per BTU
- Propane: \(\$0.0175 ÷ 2,250 = \$0.00000778\) per BTU
Perfect match!
Phase 4: Solution
Final Answer:
- Natural gas cost per cubic foot: $0.0070
- Propane cost per cubic foot: $0.0175
These selections ensure that both fuels have the same cost per BTU of usable heat energy when burned in the 90% efficient furnace.