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Chapter 1: Problem 40

. The one-liter cube in the photo has been marked off into smaller cubes, with linear dimensions one tenth those of the big one. What is the volume of each of the small cubes?(solution in the pdf version of the book)

Short Answer

Expert verified
Each small cube has a volume of 1 cubic centimeter (1 cm鲁).

Step by step solution

01

Understand the Large Cube

The large cube has a volume of 1 liter. A liter is equivalent to 1000 cubic centimeters (cm^3). Since the cube is a perfect cube, each side is the cube root of the volume: Length = \(\sqrt[3]{1000} = 10 \) cm.
02

Recognize the Dimensions of Small Cubes

Each side of the smaller cubes is one tenth the length of the large cube's sides. Therefore, each small cube has a side length of \(\frac{10}{10} = 1\) cm.
03

Calculate the Volume of Each Small Cube

Use the formula for the volume of a cube: \(V = a^3\), where \(a\) is the side length. For the small cubes, \(a = 1\) cm, so \(V = 1^3 = 1\) cubic centimeter (cm^3).

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Key Concepts

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

Cubic Centimeters
When we talk about volume in terms of cubic centimeters, we are essentially measuring how much space an object occupies. A cubic centimeter is a cube with each side measuring 1 centimeter. This small cube represents the basic unit of volume in the metric system for small objects.
  • 1 cubic centimeter ( cm鲁) is the same as the volume occupied by a cube with edges of 1 cm each.
  • It鈥檚 often used in scientific contexts to measure liquids and solids' volume.
  • The relation to other metric units, such as milliliters, is straightforward, where 1 cm鲁 equals 1 milliliter.
Being aware of this conversion is particularly handy in everyday life, especially when cooking or working with measurements in scientific experiments.
Cubic Volume
Understanding cubic volume is essential when dealing with three-dimensional spaces. Cubic volume allows us to calculate how much 3D space an object occupies. We use this measurement in various fields like shipping, architecture, and design.
  • For a cube, the volume is found using the formula: \( V = a^3 \), where \(a\) is the length of one side.
  • It's a powerful way to understand and visualize the potential capacity inside a given space.
  • Cubic volume ensures accurate measurements and is a fundamental concept in geometry.
These calculations are crucial when the exact amount of space or capacity of certain shapes or containers needs to be determined. In every practical application, from storage solutions to packaging, cubic volume offers precision and clarity.
Geometry
Geometry lays the foundation for understanding shapes and spaces. It forms the cornerstone of our ability to calculate and comprehend the universe鈥檚 physical dimensions. Here鈥檚 why geometry is vital:
  • Geometry helps us calculate areas and volumes, providing insight into how objects occupy space.
  • It builds an understanding of shapes, ranging from simple to complex, such as spheres, rectangles, and cubes.
  • Through geometry, we learn concepts like congruence, similarity, and the properties of angles and lines.
By comprehending geometry, one gains the ability to tackle real-world problems involving space, design, and rigorous logical reasoning. This essential aspect of mathematics plays a crucial role in fields like engineering, urban planning, and everyday decision-making.

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Most popular questions from this chapter

In the last century, the average age of the onset of puberty for girls has decreased by several years. Urban folklore has it that this is because of hormones fed to beef cattle, but it is more likely to be because modern girls have more body fat on the average and possibly because of estrogen-mimicking chemicals in the environment from the breakdown of pesticides. \(\mathrm{A}\) hamburger from a hormone-implanted steer has about \(0.2 \mathrm{ng}\) of estrogen (about double the amount of natural beef). A serving of peas contains about 300 ng of estrogen. An adult woman produces about \(0.5 \mathrm{mg}\) of estrogen per day (note the different unit!). (a) How many hamburgers would a girl have to eat in one day to consume as much estrogen as an adult woman's daily production? (b) How many servings of peas? (answer check available at lightandmatter.com)

In Europe, a piece of paper of the standard size, called \(\mathrm{A} 4\), is a little narrower and taller than its American counterpart. The ratio of the height to the width is the square root of 2, and this has some useful properties. For instance, if you cut an A4 sheet from left to right, you get two smaller sheets that have the same proportions. You can even buy sheets of this smaller size, and they're called A5. There is a whole series of sizes related in this way, all with the same proportions. (a) Compare an A5 sheet to an A4 in terms of area and linear size. (b) The series of paper sizes starts from an A0 sheet, which has an area of one square meter. Suppose we had a series of boxes defined in a similar way: the \(\mathrm{B} 0\) box has a volume of one cubic meter, two \(\mathrm{B} 1\) boxes fit exactly inside an \(\mathrm{B} 0 \mathrm{box}\), and so on. What would be the dimensions of a B0 box? (answer check available at lightandmatter.com)

(solution in the pdf version of the book) Compare the light-gathering powers of a 3-cm-diameter telescope and a \(30-\mathrm{cm}\) telescope.

In a computer memory chip, each bit of information (a 0 or a 1 ) is stored in a single tiny circuit etched onto the surface of a silicon chip. The circuits cover the surface of the chip like lots in a housing development. A typical chip stores \(64 \mathrm{Mb}\) (megabytes) of data, where a byte is 8 bits. Estimate (a) the area of each circuit, and (b) its linear size.

You take a trip in your spaceship to another star. Setting off, you increase your speed at a constant acceleration. Once you get half-way there, you start decelerating, at the same rate, so that by the time you get there, you have slowed down to zero speed. You see the tourist attractions, and then head home by the same method. (a) Find a formula for the time, \(T\), required for the round trip, in terms of \(d\), the distance from our sun to the star, and \(a\), the magnitude of the acceleration. Note that the acceleration is not constant over the whole trip, but the trip can be broken up into constant-acceleration parts. (b) The nearest star to the Earth (other than our own sun) is Proxima Centauri, at a distance of \(d=4 \times 10^{16} \mathrm{~m}\). Suppose you use an acceleration of \(a=10 \mathrm{~m} / \mathrm{s}^{2}\), just enough to compensate for the lack of true gravity and make you feel comfortable. How long does the round trip take, in years? (c) Using the same numbers for \(d\) and \(a\), find your maximum speed. Compare this to the speed of light, which is \(3.0 \times 10^{8}\) ? Itextup \(\\{\mathrm{m}\\} \wedge\) textup\\{s\\}. (Later in this course, you will learn that there are some new things going on in physics when one gets close to the speed of light, and that it is impossible to exceed the speed of light. For now, though, just use the simpler ideas you've learned so far.) (answer check available at lightandmatter.com)
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