Problem 84 - Algebra and Trigonometry

Question

Geometry A circular disk of radius \(R\) is cut out of paper, as shown in figure (a). Two disks of radius \(\frac{1}{2} R\) are cut out of paper and placed on top of the first disk, as in figure (b), and then four disks of radius \(\frac{1}{4} R\) are placed on these two disks, as in figure (c). Assuming that this process can be repeated indefinitely, find the total area of all the disks.

Step-by-Step Solution

Answer

The total area is \( 2 \pi R^2 \).

Step 1: Identify the Problem

We are given a series of circular disks where the radius decreases by half with each subsequent layer placed on the previous disks. Our task is to find the total area of all the disks when the process is repeated indefinitely.

Step 2: Write the Area Formula for a Circle

The area of a circle is given by the formula \( A = \pi r^2 \), where \(r\) is the radius of the circle.

Step 3: Calculate the Area of Each Set of Disks

Start with the largest disk of radius \(R\): \( \pi R^2 \). Two disks of radius \( \frac{1}{2}R \) have a combined area of \( 2 \pi \left(\frac{1}{2}R\right)^2 = \frac{1}{2} \pi R^2 \). Four disks of radius \( \frac{1}{4}R \) have a combined area of \( 4 \pi \left(\frac{1}{4}R\right)^2 = \frac{1}{4} \pi R^2 \).

Step 4: Recognize the Geometric Series

Identify that the sum of the areas forms a geometric series: \( \pi R^2 + \frac{1}{2} \pi R^2 + \frac{1}{4} \pi R^2 + \ldots \). This is a geometric series where the first term \( a = \pi R^2 \) and the common ratio \( r = \frac{1}{2} \).

Step 5: Sum the Infinite Geometric Series

The sum of an infinite geometric series \( S = \frac{a}{1-r} \). Substituting the known values gives \( S = \frac{\pi R^2}{1 - \frac{1}{2}} = 2 \pi R^2 \).

Step 6: State the Total Area

The total area of all the disks when the process is repeated indefinitely is \( 2 \pi R^2 \).

Key Concepts

Area of a Circle

The area of a circle is calculated using the formula \( A = \pi r^2 \). This formula tells you how much space is inside a circle. Here, \(r\) stands for the radius of the circle, and \(\pi\) (pi) is a special mathematical constant approximately equal to 3.14159.
To understand it better:

• "Radius" is the distance from the center of the circle to any point on its boundary.
• The "square" \( (r^2) \) means you multiply the radius by itself.
• Multiplying \(\pi\) with \(r^2\) gives the total area inside the circle.

So, if you have a circle with a radius \(R\), its area is \(\pi R^2\). Knowing how to calculate the area of a circle is essential for solving problems involving circular shapes.

Infinite Series

An infinite series is a sum of terms that goes on forever. In our exercise, each layer of circles adds to the previous total area, and the process continues indefinitely.
The sequence we are dealing with is a geometric series because the ratio between the consecutive terms is constant. This is important because it allows us to use specific formulas to find the total sum, even if it technically never ends.
In a geometric series, the first term is called \(a\) and the common ratio \(r\) is the factor that you multiply each term by to get the next term. For our problem, the first term \(a\) is the area of the largest circle \(\pi R^2\), and the common ratio \(r\) is \(\frac{1}{2}\).
The solution to this problem involves adding up an infinite number of circular areas to understand how they combine in this never-ending process. Using the formula for the sum of an infinite geometric series \(S = \frac{a}{1-r}\), we can find that even though there are infinite circles, the total area can still be finite.

Radius

The radius is a fundamental aspect of any circle, representing its size. It is the distance from the center of the circle to its edge. In mathematical terms, a radius is half the length of the diameter, which spans the entire circle through the center point.
Understanding the radius is crucial because it is used in formulas to calculate both the circumference and the area of a circle.
Here's a quick summary:

• For a circle, the formula for circumference is \(C = 2\pi r\).
• For area, it's \(A = \pi r^2\).

In our exercise, the radius of each subsequent disk is a fraction of the previous one (e.g., \(\frac{1}{2} R\), \(\frac{1}{4} R\)). This pattern is key in defining the progression of areas in the geometric series calculations.

Geometric Progression

In mathematics, a geometric progression is a sequence where each term after the first is found by multiplying the previous term by a fixed number called the common ratio. Understanding this concept is essential for solving our exercise where the areas of disks form such a progression.
For example, in our problem:

• The first disk's area is \(\pi R^2\).
• The next set of disks has an area that is \(\frac{1}{2}\) of the initial area.
• The third set's area is \(\frac{1}{4}\) of the initial area, and so on.

The Common Ratio:

• Here, the common ratio \(r\) is \(\frac{1}{2}\). This means each layer's total area is half of the one before it.

When you identify a sequence as a geometric progression, it becomes easier to calculate sums, especially when dealing with infinite terms. You can use the formula for the sum of an infinite geometric series, which hugely simplifies finding the total area of all disks in this exercise.