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 3.536 meters.

Step 1: Understanding the Formula

The formula to find the circumference of a circle when you have the radius is: \( C = 2 \pi r \). Here, \( C \) represents the circumference and \( r \) represents the radius of the circle.

Step 2: Identify the Given Value

We are given that the radius \( r = 0.563 \) meters. This is the value we will use in our formula.

Step 3: Substitute the Radius into the Formula

Plug the given radius into the formula for circumference: \( C = 2 \pi \times 0.563 \).

Step 4: Calculate the Circumference

Calculate the value by performing the multiplication: \( C = 2 \times 3.14159 \times 0.563 \approx 3.536 \text{ meters} \).

Step 5: Round Off the Answer

The calculated circumference is approximately 3.536 meters. Depending on required precision, you can round this number if necessary, but it is reasonable to use two or three decimal places as it reflects precision in the context of practical measurements.

Key Concepts

Circle Radius

The radius of a circle is the distance from the center of the circle to any point on its edge. It is a crucial measurement as it directly influences the size of the circle and is fundamental in calculating other elements like the circumference and area. In our problem, the radius is given as 0.563 meters.
Knowing the radius allows us to derive a host of other properties of the circle using simple mathematics.

• The radius is half of the diameter, so if you know one, you can easily find the other with the relation: diameter = 2 × radius.
• The radius is a constant measure for all points on the circle.
• It is one of the most essential input values for circle-related formulas.

Comprehending the radius helps us to unlock other geometrical elements and equations tied to the circle.

Circle Formulas

Circle formulas are mathematical expressions used to calculate different properties of a circle. One of the most common formulas is that for the circumference, which requires either the diameter or radius of the circle.
The circumference is essentially the distance around the circle. For a circle with radius, the formula is given by:
\[ C = 2\pi r \]
Where

• \( C \) is the circumference
• \( \pi \) is approximately 3.14159 (though often simply referred to as 3.14 or even represented by the fraction \( \frac{22}{7} \) in basic calculations).
• \( r \) is the radius of the circle.

By substituting the known radius into this formula, we easily compute the circumference. These formulas provide a powerful way to understand and quantify the characteristics of circles in geometry.

Mathematics Problem Solving

Mathematics problem solving, particularly in geometry, involves several steps that help us reach the desired solution systematically. To solve our given problem of finding the circumference from the radius, following structured steps ensures accuracy:

• **Understanding the Problem:** Clearly identify what is being asked and what values are given.
• **Applying the Right Formula:** Recognize and use the appropriate mathematical formula based on the given information.
• **Substitution and Calculation:** Substitute the known values into the formula and perform the necessary arithmetic calculations, ensuring precision.
• **Verification and Rounding Off:** Verify whether the calculated value meets the problem requirements, rounding off if necessary for practical measurements.

By approaching problems with a clear method, we not only find solutions but also deepen our understanding of mathematical concepts and improve problem-solving skills.