Problem 50 - Algebra 2

Question

Music. A CD with diameter 12 \(\mathrm{cm}\) spins in a CD player. Calculate how much farther a point on the outside edge of the CD travels in one revolution than a point 1 \(\mathrm{cm}\) closer to the center of the CD.

Step-by-Step Solution

Answer

The point on the outside edge of the CD travels approximately \(2\pi\) cm farther in one revolution than a point 1 cm closer to the center of the CD.

Step 1: Identify the Radius of Both Points

Firstly, we need to find the radius of both circles. Taking into account that the diameter of the CD is 12 cm, the radius of the outside edge of the CD is \(\frac{12}{2} = 6\) cm. The radius for the point 1 cm closer to the center will therefore be \(6 - 1 = 5\) cm.

Step 2: Calculate the Circumference of Each Point

Next, we calculate the circumference of the circles of the two points. The formula for the circumference of a circle is \(C = 2 \pi r\). Therefore, the circumference of the outer edge is \(C_1 = 2 \pi \cdot 6\) and the circumference of the point 1 cm closer to the center is \(C_2 = 2 \pi \cdot 5\).

Step 3: Find the Difference in Distances

The final step is to find the difference in distances traveled in one revolution. This is done by subtracting the circumference of point 1 cm closer to the center from the circumference of the outer point: \(C_1 - C_2 = 2 \pi \cdot 6 - 2 \pi \cdot 5\).

Key Concepts

Circumference Calculation

To understand how far a point on the edge of a CD travels during one complete revolution, it's essential to grasp the concept of circumference. The circumference of a circle is the total distance around it, much like the perimeter of a polygon.
For circles, this is calculated using the formula:

• \( C = 2 \pi r \)

where \( r \) is the radius of the circle. The radius is the distance from the circle's center to any point on its edge. By substituting the radius into this formula, you can find out the total length traveled by a point on the circle's edge in one complete revolution.
In the case of a 12 cm diameter CD, the calculation would be specific to the outer edge. Using the formula, you can determine the distance the outermost point on the circle would move.

Radius of a Circle

The radius is a crucial part of any circle-related calculation. It is half of the diameter, which is the length of a straight line passing from one side of a circle to the other, through the center. So, if you know the diameter, finding the radius is straightforward:

• \( r = \frac{d}{2} \)

For instance, a CD with a 12 cm diameter has a radius of 6 cm.
Knowing the radius helps in determining other circular measurements like the circumference.
Furthermore, in our exercise, when you want to find the radius of a point on the CD that's 1 cm closer to the center, simply subtract 1 cm from the radius of the outer edge. This results in a 5 cm radius for that particular point.

Difference in Distances

When two points on a CD spin around, they cover different distances based on their respective circumferences. The outermost point will travel further than a point closer to the center. To find this difference, you need to calculate their circumferences first, as explained in the earlier sections.
For example:

• The circumference of the outer edge: \( 2 \pi \times 6 \)
• The circumference of the point 1 cm closer: \( 2 \pi \times 5 \)

Then, subtract the smaller circumference from the larger to find how much farther the outer point travels:

• \( (2 \pi \times 6) - (2 \pi \times 5) \)

This simple subtraction helps to determine the additional distance covered by the outside edge compared to the point closer to the center, after one full revolution.