Problem 10 - Basic Technical Mathematics with Calculus

Question

Find the circumference of the circle with the given radius or diameter. $$r=0.563 \mathrm{m}$$

Step-by-Step Solution

Answer

The circumference is approximately 3.5368 meters.

Step 1: Recall the Formula for Circumference

The circumference of a circle is calculated using the formula \( C = 2 \pi r \), where \( C \) is the circumference and \( r \) is the radius of the circle.

Step 2: Substitute the Radius into the Formula

Insert the given radius \( r = 0.563 \mathrm{m} \) into the formula from Step 1. This gives us:\[ C = 2 \pi (0.563) \]

Step 3: Calculate the Circumference

Calculate the value of \( C = 2 \pi (0.563) \). Using \( \pi \approx 3.14159 \), the calculation becomes:\[ C = 2 \times 3.14159 \times 0.563 \approx 3.5368 \mathrm{m} \]

Step 4: Write the Final Answer

The calculated circumference of the circle, rounded to the nearest four decimal places, is approximately \( 3.5368 \mathrm{m} \).

Key Concepts

radius of a circle

The radius of a circle is a key concept in geometry. It's the distance from the center of the circle to any point on its edge. Understanding the radius is crucial because it acts as a building block for other calculations, like finding the circle's circumference or area. Here’s what is interesting about the radius:

• It's always equal no matter where you measure from the center to the edge.
• If you know the radius, you can calculate the diameter by doubling it (since the diameter is the distance across the circle passing through the center).
• The radius is often given in problems because it's a straightforward measure that simplifies calculations.

For example, if a problem tells you the circle has a radius of 0.563 meters, like in our original exercise, you can directly use that value for further calculations.

circumference formula

When you need to find the distance around a circle, you use the circumference formula. The standard formula for circumference is:\[ C = 2 \pi r \] This formula tells us two things:

• We need to multiply two times the product of \( \pi \) and the radius.
• The formula demonstrates that circumference directly depends on the radius. A larger radius means a larger circle, thus more distance around it.

In the exercise, we use the given radius of 0.563 meters and substitute it into the formula, giving us \( C = 2 \pi (0.563) \). This step sets the stage for calculating the true circumference.

calculation with pi

\( \pi \), a fascinating number, is central to calculations involving circles. Representing the ratio of the circumference of any circle to its diameter, it's approximately equal to 3.14159, although it can go on forever as a non-repeating decimal. When calculating the circumference:

• We use \( \pi \) to accurately scale the circle’s radius to find that circle's circumference.
• Despite often being rounded to 3.14 or 22/7 in simpler computations, more precise problems will use a longer version such as 3.14159 or even more exact values depending on the needs.

This means when you perform the calculation \( C = 2 \pi (0.563) \), you multiply 2 with 3.14159 and then with 0.563, resulting in an approximate circumference of 3.5368 meters, as calculated in our example. This shows just how critical \( \pi \) is for precision in measurement.