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Chapter 1: Problem 20

What is the volume in cubic metres of a cube of side \(2.8 \mathrm{cm} ?\)

Short Answer

Expert verified
0.000021952 cubic meters

Step by step solution

01

- Convert the side length to meters

The side length of the cube is given in centimeters (cm). Convert it to meters (m) since the final volume is required in cubic meters.Given: Side length = 2.8 cm.Conversion factor: 1 meter = 100 centimeters.Calculate: \[ \text{Side length in meters} = \frac{2.8}{100} = 0.028 \text{ meters} \]
02

- Apply the volume formula for a cube

The volume of a cube is given by the formula \[ V = a^3 \],where \( a \) is the side length of the cube.Given: \( a = 0.028 \text{ meters} \)Calculate: \[ V = (0.028)^3 \]
03

- Calculate the cube of the side length

Now, calculate the cube of 0.028 meters:\[ V = 0.028 \times 0.028 \times 0.028 = 0.000021952 \text{ cubic meters} \]
04

- State the final volume

The volume of the cube is:\[ V = 0.000021952 \text{ cubic meters} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conversion of Units
When solving problems involving volume, it鈥檚 essential to use consistent units for accuracy. Here, we needed to convert the side length from centimeters to meters because the volume should be in cubic meters.

Remember, there are 100 centimeters in a meter. Therefore, to convert centimeters to meters, divide the number of centimeters by 100.

For instance, given a side length of 2.8 cm, the conversion to meters is done as follows:

\[ \frac{2.8 \text{ cm}}{100} = 0.028 \text{ m} \]

By using consistent units, you ensure that your volume calculation is correct and in the desired unit of cubic meters.
Cube Volume Formula
Calculating the volume of a cube involves a straightforward formula.

The volume \[ V \] of a cube with side length \[ a \] is given by: \[ V = a^3 \]

This formula states that the volume is the cube of the side length. Meaning, you multiply the side length by itself three times (length 脳 width 脳 height, which are all the same in a cube).

For the given problem, the side length \[ a = 0.028 \text{ meters} \]. Applying the formula: \[ V = (0.028)^3 = 0.028 \times 0.028 \times 0.028 = 0.000021952 \text{ cubic meters} \]

By understanding this base formula, you can then adjust and apply it to different units or dimensions as needed.
Metric System
The metric system is a decimal-based system of measurement used around the world.

Key units in the metric system include meters for length, kilograms for mass, and liters for volume. One vital aspect of the metric system is its use of prefixes, like centi- (one hundredth), milli- (one thousandth), and kilo- (one thousand).

For example:
  • 1 meter = 100 centimeters
  • 1 meter = 1000 millimeters
  • 1 kilometer = 1000 meters


This standardization makes conversions straightforward. For instance, converting from centimeters to meters involves dividing by 100.

Understanding the metric system is crucial for science, engineering, and everyday life, simplifying measurements and making calculations universally comprehensible.

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Most popular questions from this chapter

In a similar experiment, the following data was collected for current and voltage: \((V, I)=\) ((0.1,27),(0.2,44),(0.3,60),(0.4,78)} with an uncertainty of \(\pm 4 \mathrm{mA}\) in the current. Plot the current versus the voltage and draw the line of best fit. Can it be claimed that the line passes through the origin?

The kinetic energy of a mass \(m\) is given by \(\frac{1}{2} m v^{2} .\) If the speed \(v\) is doubled, by what factor does the energy change?

The kinetic energy \(\left(E_{k}=\frac{1}{2} m v^{2}\right)\) of a body doubles. By what factor did the speed increase?

The image of an object a distance a from a lens is formed at a distance \(b\) from the lens, where a and \(b\) are related through the equation \(\frac{1}{2}+\frac{1}{b}=\frac{1}{f}\) and \(f\) is the (constant) focal length of the lens.(a) If a set of data for \(a\) and \(b\) is collected, how should it be plotted in order to give a straight line? (b) How can the focal length of the lens be measured from the graph?

In a certain collision, the momentum vector of a particle changes direction but not magnitude. Let \(\vec{p}\) be the momentum vector of a particle suffering an elastic collision and changing direction by \(30^{\circ} .\) Find, in terms of \(p(=|\vec{p}|),\) the magnitude of the vector representing the change in the momentum vector.
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