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Chapter 1: Problem 39

The central portion of a CD is taken up by the hole and some surrounding clear plastic, and this area is unavailable for storing data. The radius of the central circle is about \(35 \%\) of the outer radius of the data-storing area. What percentage of the CD's area is therefore lost? (answer check available at lightandmatter.com)

Short Answer

Expert verified
12.25% of the CD's area is unavailable for storing data.

Step by step solution

01

Understand the Problem

You're asked to find the percentage of the CD's area that is unavailable for storing data because of the central hole and surrounding clear plastic. The central portion has a radius of 35% of the data-storing area's outer radius.
02

Define the Variables

Let the radius of the outer data-storing area be denoted as \( R \). Therefore, the radius of the central portion, which includes the hole and clear plastic, is \( 0.35R \).
03

Calculate the Area of the Full CD

The area of the full circle (or CD) is calculated using the formula for the area of a circle, \( A = \pi R^2 \). Here, \( R \) is the radius of the CD.
04

Calculate the Area of the Central Portion

The area of the central portion (which is not used for data) can be found using the formula for the area of a circle: \( A_{central} = \pi (0.35R)^2 \). Simplifying this gives \( A_{central} = \, \pi \times 0.1225R^2 \).
05

Determine the Fraction of the Total Area

To find the fraction of the CD's area that is unavailable, divide the central area's area by the full CD's area: \( \text{Unavailable fraction} = \frac{A_{central}}{A} = \frac{0.1225 \pi R^2}{\pi R^2} = 0.1225 \).
06

Convert Fraction to Percentage

Multiply the fraction by 100 to convert it to a percentage: \( \text{Unavailable percentage} = 0.1225 \times 100 = 12.25\% \). Thus, 12.25% of the CD's surface area is lost.

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Key Concepts

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

Circle Area Calculation
Understanding how to calculate the area of a circle is a foundational skill in geometry. A circle's area is determined by the formula \( A = \pi R^2 \), where \( A \) represents the area, \( R \) is the radius, and \( \pi \) is a constant approximately equal to 3.14159. The radius is the distance from the center of the circle to any point on its edge.
  • Step-by-step: To calculate, you square the radius \((R^2)\) and multiply it by \( \pi \). This gives the full area that the circle occupies in a plane.
  • Importance: Knowing this formula is essential for various applications, such as calculating land areas, understanding sizes, and in design industries.

When solving problems involving circular areas, take note of any changes to dimensions that might affect the total area occupied by circles.
Percentage of Area
Determining the percentage of a certain area is pivotal, especially in understanding how much of a space is occupied or available. This concept can be seen as a ratio of areas. In the context of the CD problem, we wanted to find what percentage of the total area is not used for storing data.
  • Step-by-step: First, you determine the area that's unavailable by using the area formula. Then, you compare this to the total area of the CD. The fraction you get from this division tells you the proportion of the area lost.
  • Conversion: Converting this fraction into a percentage involves multiplying by 100. This makes it easier to understand and communicate the proportion in everyday terms.

This conversion process helps in practical applications where quick, comprehensible insights are needed, such as planning or optimizations.
Radius and Diameter Relationships
The relationship between a circle鈥檚 radius and diameter is simple yet crucial. The diameter is twice the radius, represented by the formula \( D = 2R \), where \( D \) is the diameter and \( R \) is the radius.
  • Key understanding: Knowing this relationship helps in various geometry problems, allowing us to convert between measurements easily. For instance, if you know the diameter, you can quickly find the radius by halving the diameter.
  • Application: In our CD problem, understanding the relationship between the radius of the central portion and the outer radius is crucial for correctly calculating the areas involved.

Recognizing these relationships simplifies calculations and enhances efficiency in both academic and practical scenarios, from designing wheels to crafting circular tables.

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Most popular questions from this chapter

A hunting dog's nose has about 10 square inches of active surface. How is this possible, since the dog's nose is only about 1 in \(\times 1\) in \(\times 1\) in \(=1 \mathrm{in}^{3}\) ? After all, 10 is greater than 1, so how can it fit?

The speed of light is \(3.0 \times 10^{8} \mathrm{~m} / \mathrm{s}\). Convert this to furlongs per fortnight. A furlong is 220 yards, and a fortnight is 14 days. An inch is \(2.54 \mathrm{~cm}\).(answer check available at lightandmatter.com)

According to folklore, every time you take a breath, you are inhaling some of the atoms exhaled in Caesar's last words. Is this true? If so, how many?

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)

You are looking into a deep well. It is dark, and you cannot see the bottom. You want to find out how deep it is, so you drop a rock in, and you hear a splash \(3.0\) seconds later. How deep is the well? (answer check available at lightandmatter.com)
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