Question

Gasoline's density is about \(0.65 \mathrm{~g} / \mathrm{mL}\). How much does \(34 \mathrm{~L}\) (approximately 18 gallons) weigh in kilograms? In pounds?

Short answer

34 liters of gasoline weighs approximately 22.1 kilograms or about 48.7 pounds.

Step-by-step solution

Convert Liters to Milliliters

There are 1000 milliliters (mL) in a liter (L). To convert 34 liters to milliliters, multiply the volume in liters by 1000 as milliliters are smaller units.

Calculate the Mass in Grams

Now that we have the volume in milliliters, use the density of gasoline to find the mass in grams. The density is 0.65 grams per milliliter, so multiply the volume in milliliters by the density.

Convert the Mass to Kilograms

There are 1000 grams in one kilogram. To convert the mass from grams to kilograms, divide the mass in grams by 1000.

Convert Kilograms to Pounds

To convert kilograms to pounds, use the conversion factor that 1 kilogram equals approximately 2.20462 pounds. Multiply the mass in kilograms by this conversion factor to get the mass in pounds.

Key concepts

Unit Conversion

Understanding unit conversion is crucial when dealing with measurements in various units. For instance, the problem given involves converting the volume of gasoline from liters to milliliters, and then the mass from grams to kilograms and pounds. This process is vital because we often measure liquids in liters, but density is typically expressed in grams per milliliter. Therefore, the initial conversion allows us to use the density effectively.

When converting units, it's essential to know the conversion factors, such as 1 liter equalling 1000 milliliters or 1 kilogram equalling 2.20462 pounds. This information is the key to transforming one unit into another, which allows for accurate calculations across different systems of measurement, like metric to imperial. Through practice, these conversions become second nature, allowing for quick and efficient problem-solving in various scientific contexts.

To improve grasp of unit conversions:

• Memorize common conversion factors.
• Always check units before and after conversion to ensure accuracy.
• Utilize unit labels in calculations to track which conversions have been made.

Mass-Volume Relationship

The mass-volume relationship is the backbone of many scientific calculations, particularly when dealing with substances like liquids and gases. In our exercise, density serves as the conversion factor between the volume of the gasoline in milliliters and its mass in grams.

Density is defined as mass per unit volume, commonly expressed as grams per milliliter (\text{g/mL}). Knowing the substance's density allows one to calculate the mass if the volume is known and vice versa. This relationship is expressed in the formula: \[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \. \] By rearranging this formula, we can solve for mass: \[ \text{Mass} = \text{Density} \times \text{Volume}. \] In the given problem, multiplying the volume of gasoline by its density gives us the total mass of the gasoline.

Here are some tips for mastering mass-volume relationships:

• Ensure consistency of units across mass and volume.
• Understand that density is a specific property of each material and varies accordingly.
• Practice calculating one variable when given the other two to reinforce the concept.

Dimensional Analysis

Essence of Dimensional Analysis

Dimensional analysis, also known as the factor-label method, is an immensely powerful tool in solving problems that involve unit conversions and relationships between different physical quantities. By treating units as algebraic quantities, we can cancel them systematically to achieve the desired units for our answer.

Application in Problem Solving

Applying dimensional analysis to the given problem, we set up conversion factors with units positioned so they cancel out, leaving us with the desired unit. For example, to convert 34 liters to milliliters, we set up a conversion factor where liters will cancel out, as such: \[ 34 \, \text{L} \times \frac{1000 \, \text{mL}}{1 \, \text{L}} = \text{Mass in milliliters}. \]

This method applies to all subsequent steps, ensuring that every conversion from grams to kilograms, and then to pounds, is accurate. Dimensional analysis is a universal technique applicable not only for scientific calculations but also in daily life, such as when cooking or shopping.

To effectively use dimensional analysis:

• Write down the units with numbers and treat them as algebraic entities.
• Align conversion factors so units cancel out, step by step, until you reach the desired unit.
• Review conversion factors regularly to become familiar with their use in different contexts.