Convert 1.09 g/mL to lbs/gal using dimensional analysis please show steps
The Correct Answer and Explanation is :
To convert 1.09 g/mL to lbs/gal using dimensional analysis, we will follow a step-by-step approach using known conversion factors between grams and pounds, milliliters and gallons.
Step 1: Write the given value and set up the conversion
You are starting with a density of 1.09 grams per milliliter (g/mL). To convert to pounds per gallon (lbs/gal), we need to convert both grams to pounds and milliliters to gallons.
The dimensional analysis setup will look like this:
[
1.09 \, \text{g/mL} \times \left( \frac{\text{1 lb}}{453.592 \, \text{g}} \right) \times \left( \frac{\text{3.78541 L}}{1 \, \text{gal}} \right) \times \left( \frac{1000 \, \text{mL}}{\text{1 L}} \right)
]
Step 2: Conversion factors
- ( 1 \, \text{lb} = 453.592 \, \text{g} ) (Convert grams to pounds)
- ( 1 \, \text{gal} = 3.78541 \, \text{L} ) (Convert gallons to liters)
- ( 1 \, \text{L} = 1000 \, \text{mL} ) (Convert milliliters to liters)
Step 3: Perform the calculations
- Convert grams to pounds:
[
1.09 \, \text{g} \times \left( \frac{\text{1 lb}}{453.592 \, \text{g}} \right) = 0.0024 \, \text{lb}
] - Convert milliliters to gallons:
[
1 \, \text{mL} \times \left( \frac{\text{3.78541 L}}{1000 \, \text{mL}} \right) = 0.00378541 \, \text{L}
]
[
0.00378541 \, \text{L} \times \left( \frac{\text{1 gal}}{3.78541 \, \text{L}} \right) = 0.00026417 \, \text{gal}
] - Final result:
[
\text{Density in lbs/gal} = 0.0024 \, \text{lb/mL} \times \left( \frac{1}{0.00026417 \, \text{gal/mL}} \right) = 8.98 \, \text{lbs/gal}
]
Conclusion
1.09 g/mL is approximately 8.98 lbs/gal.
Explanation:
This problem involves converting from metric units (grams and milliliters) to imperial units (pounds and gallons). Grams are a unit of mass, while milliliters measure volume. Using dimensional analysis, we set up a series of conversion factors to cancel out the units of grams and milliliters, replacing them with pounds and gallons.
The conversions are straightforward:
- Grams to pounds using the factor 1 lb = 453.592 g.
- Milliliters to gallons using 1 liter = 1000 milliliters and 1 gallon = 3.78541 liters.
After performing the calculations, we arrive at the final density of 8.98 pounds per gallon. This demonstrates how dimensional analysis allows you to convert between different units systematically while ensuring the units are consistent across the conversion process.