Question

Perform the following conversions. Note that you will have to convert units in both the numerator and the denominator. a) \(3.88 \times 10^{2} \mathrm{~mm} / \mathrm{s}\) to kilometers/hour b) \(1.004 \mathrm{~kg} / \mathrm{L}\) to grams/milliliter

Short answer

a) 1.3968 km/h b) 1.004 g/mL

Step-by-step solution

Convert Millimeters to Kilometers

To convert from millimeters to kilometers, we need to understand the conversion factor. 1 kilometer is equal to 1,000,000 millimeters. So, to convert 3.88 \times 10^2 mm to kilometers, divide by 1,000,000. \[3.88 \times 10^2 \text{ mm} \times \frac{1 \text{ km}}{1,000,000 \text{ mm}} = 3.88 \times 10^{-4} \text{ km}\]

Convert Seconds to Hours

Since we are converting the rate from millimeters per second to kilometers per hour, the time units also need conversion. 1 hour is equal to 3600 seconds. Multiply the conversion into kilometers by this factor. \[3.88 \times 10^{-4} \text{ km/s} \times 3600 \text{ s/h} = 1.3968 \text{ km/h}\]

Calculate Kilometers per Hour

After converting both components (distance and time), we find that \(3.88 \times 10^2 \text{ mm/s}\) is equal to \(1.3968 \text{ km/h}\).

Convert Kilograms to Grams

To convert from kilograms to grams, use the conversion factor where 1 kilogram equals 1,000 grams. Multiply \(1.004 \text{ kg}\) by this factor. \[1.004 \text{ kg/L} \times 1,000 \text{ g/kg} = 1,004 \text{ g/L}\]

Convert Liters to Milliliters

Next, convert liters to milliliters. Since 1 liter is equal to 1,000 milliliters, divide the gram per liter value by 1,000 to get grams per milliliter. \[1,004 \text{ g/L} \times \frac{1 \text{ L}}{1,000 \text{ mL}} = 1.004 \text{ g/mL}\]

Calculate Grams per Milliliter

After performing the conversion, we find that \(1.004 \text{ kg/L}\) is equal to \(1.004 \text{ g/mL}\).

Key concepts

Metric System Conversions

The metric system is a decimal-based system of measurement used worldwide. It is especially useful for scientific and technical purposes because it uses a series of units that are scalable by powers of ten. This makes conversions within the system predictable and manageable.

• Length Conversions: For length, the metric system uses units like millimeters (mm), centimeters (cm), meters (m), and kilometers (km). Moving from smaller to larger units involves division by powers of ten, while moving from larger to smaller units involves multiplication. For instance, converting millimeters to kilometers involves dividing by a million, as there are 1,000 mm in a meter and 1,000 meters in a kilometer.
• Volume and Mass Conversions: For volume, we often encounter liters (L) and milliliters (mL), where 1 L equals 1,000 mL. For mass, kilograms (kg) and grams (g) are the standard units, with 1 kg equaling 1,000 g. This straightforward scaling by ten makes calculations simple and logical.

Understanding these conversions becomes invaluable, as in our given exercise where both the numerator and denominator units require conversion. Having a solid grasp of these principles is crucial for accurate and efficient computations.

Density Conversion

Density is a measure of how much mass is contained in a given volume. It is commonly expressed in units such as kilograms per liter (kg/L) or grams per milliliter (g/mL). Conversion between these units follows the basic principles of metric conversions.
For instance, converting from kilograms per liter to grams per milliliter involves two main steps:

• Convert Mass: Since 1 kilogram equals 1,000 grams, multiplying the density in kg/L by 1,000 gives the density in g/L.
• Convert Volume: Next, since 1 liter equals 1,000 milliliters, dividing the result by 1,000 converts the volume to the desired mL unit.

This is evident in the straightforward solution for the exercise where 1.004 kg/L was converted to 1.004 g/mL following these principles. Understanding density conversions is essential for various fields such as chemistry, physics, and engineering, where precise measurements are crucial.

Speed Conversion

Converting speed involves changes in both distance and time units. Often, speeds are provided in units like meters per second (m/s) or kilometers per hour (km/h). To convert these, careful attention to both components is required.
In our example, the conversion from millimeters per second (mm/s) to kilometers per hour (km/h) necessitates these fundamental steps:

• Convert Distance Units: Here, mm is converted to km by dividing by 1,000,000, since there are 1,000 mm in a meter and 1,000 meters in a km.
• Convert Time Units: Convert seconds to hours by multiplying by 3600, as there are 3600 seconds in an hour.

This ensures each part of the speed unit is correctly transformed, resulting in an accurate overall speed conversion. Mastery of speed conversions allows you to interpret and communicate measurements effectively in various contexts, from everyday activities to complex engineering calculations.