Question

What should be the volume of the milk (in \(\mathrm{m}^{3}\) ) which measures \(5 \mathrm{~L} ?\) (a) \(5 \times 10^{-3} \mathrm{~m}^{3}\) (b) \(5 \times 10^{3} \mathrm{~m}^{3}\) (c) \(5 \times 10000 \mathrm{~m}^{3}\) (d) \(5 \times 10^{6} \mathrm{~m}^{3}\)

Short answer

5 x 10^{-3} \mathrm{~m}^{3}

Step-by-step solution

Understand the conversion from liters to cubic meters

1 liter is equivalent to 0.001 cubic meters (m^3), because 1 liter is the same as 1 cubic decimeter (dm^3) and there are 1000 cubic decimeters in a cubic meter.

Convert liters to cubic meters

To convert 5 liters to cubic meters, multiply the volume in liters by the conversion factor from Step 1. So, 5 L = 5 * 0.001 m^3 = 0.005 m^3.

Write the volume in scientific notation

The volume in cubic meters can be expressed in scientific notation as 5 x 10^(-3) m^3 to make it consistent with the options given.

Key concepts

Liters to Cubic Meters

Understanding how to convert from liters to cubic meters is essential when working with volume measurements in different systems of units. A liter (L) is a metric unit of volume commonly used for liquids, such as milk in our exercise. It's important to note that one liter is equal to one cubic decimeter (dm³), and since there are 1,000 cubic decimeters in one cubic meter (m³), the conversion is straightforward: 1 L = 0.001 m³.

To apply this to our problem, where we need to determine the volume of 5 liters of milk in cubic meters, we multiply the volume in liters by the conversion factor. Thus, the calculation is performed as follows:\[ 5 L \times 0.001 \frac{m^3}{L} = 0.005 m^3 \].

It's crucial for students to be comfortable with this step because converting between liters and cubic meters is a common task in various scientific and engineering fields. Remember, always multiply the amount in liters by the conversion factor (0.001) to switch to cubic meters.

Scientific Notation

Scientific notation is a method of expressing numbers that are too big or too small to be conveniently written in decimal form. It is particularly useful in science and engineering to simplify numbers and make calculations easier. When numbers are written in scientific notation, they are expressed as a product of two numbers: a coefficient and a power of 10. The format generally looks like \( a \times 10^n \), where \( a \) is the coefficient that must be greater than or equal to 1 and less than 10, and \( n \) is an integer representing the power of 10.

In the context of our exercise, the volume of milk, 0.005 m³, is converted to scientific notation for standardization and ease of understanding as follows:\[ 0.005 m^3 = 5 \times 10^{-3} m^3 \].

The power of 10, in this case, is -3 since we have to move the decimal place three positions to the left to get from 0.005 to 5. By mastering scientific notation, students can work more efficiently with very large or very small values, and communicate findings with greater precision.

Dimensional Analysis

Dimensional analysis, also known as factor-label method or unit conversion method, is a powerful tool used in solving problems involving the conversion of measurement units. This technique ensures that the final answer of a calculation is presented in the correct units by multiplying the initial quantity and units by conversion factors that cancel out the undesired units, leaving only the desired ones.

For example, in the case of converting 5 liters to cubic meters, dimensional analysis involves:\[ 5 \text{ L} \times \frac{1 \text{ m}^3}{1000 \text{ L}} \].

The liters cancel out, and we are left with the volume in cubic meters. Each conversion factor is essentially a ratio equal to one (since 1 m³ = 1000 L), so multiplying by it does not change the value, just the units. Dimensional analysis is not only limited to volume conversions; it's applicable to any type of unit conversion and is an indispensable skill in the sciences.