Problem 9 - Basic Technical Mathematics with Calculus

Question

Find the circumference of the circle with the given radius or diameter. $$r=275 \mathrm{ft}$$

Step-by-Step Solution

Answer

The circumference is approximately 1728.96 ft.

Step 1: Understand the Formula

The circumference of a circle can be calculated using the formula \( C = 2\pi r \), where \( C \) is the circumference and \( r \) is the radius of the circle. It is essential to identify that you have the radius given in the problem.

Step 2: Substitute the Given Radius

Given the radius \( r = 275 \text{ ft} \), substitute this value into the circumference formula to calculate it. This gives us the equation: \( C = 2 \pi \times 275 \).

Step 3: Calculate the Expression

Now, calculate \( C = 2 \times 3.14159 \times 275 \). This step involves multiplying these three numbers together to get the circumference of the circle.

Step 4: Compute the Final Result

After performing the multiplication, the calculation becomes \( C \approx 1728.96 \text{ ft} \), rounding to two decimal places, which is the circumference of the circle.

Key Concepts

Mathematical Formulas

Mathematical formulas are the backbone of solving geometry problems like finding the circumference of a circle. A formula provides a reliable method to calculate a specific result. In our exercise, the formula used is the circumference formula: \( C = 2\pi r \). This powerful mathematical tool allows us to find the distance around the circle, known as the circumference, simply by knowing the radius. The formula is derived from the relationship between the radius, the circle's size, and the number \( \pi \), a constant approximately equal to 3.14159. Remember, using correct formulas is crucial in solving geometric problems efficiently. It's like having a map that guides you to the solution.

Radius of a Circle

The radius of a circle is a fundamental element in geometric calculations. It is the distance from the center of the circle to any point on its circumference. Understanding this concept is essential because it is used in many formulas, such as the circumference and area of a circle.
The radius is always consistent for any point you measure from the center to the edge, making it a reliable measure of the circle's size.

• If you know the radius, you can find other properties of the circle, like its circumference.
• The radius is always half of the diameter of the circle.

In our problem, the radius provided is 275 feet, which is what we use to find the circumference accurately. Keep in mind that the radius has a significant role in determining the size and scale of the circle, influencing how we approach the problem.

Geometric Calculations

Geometric calculations involve using known values and formulas to determine unknown characteristics of shapes. It's like solving a puzzle with numbers and operations. In our exercise, the geometric calculation is about finding the circumference of a circle using its radius.
Here are the key steps involved:

• Identify: Recognize that the radius is given.
• Substitute: Plug the radius value into the formula \( C = 2\pi r \).
• Calculate: Perform the arithmetic to find the result.

By understanding how these steps work together, students can solve various geometry problems efficiently. Practicing these calculations not only sharpens math skills but also enhances problem-solving capabilities. Remember, accuracy in each step is vital to arriving at the correct solution.