Problem 49 - Prealgebra

Question

Solve each of the following word problems. Note that not all of the problems are solved by simply multiplying the numbers in the problems. Many of the problems involve addition and subtraction as well as multiplication. This Google Earth image shows an aerial view of a crop circle found near Wroughton, England. If the crop circle has a radius of 59.13 meters, what is its circumference? Use the approximation 3.14 for \(\pi\). Round to the nearest hundredth. (IMAGE CANNOT COPY)

Step-by-Step Solution

Answer

The circumference of the crop circle is approximately 370.27 meters.

Step 1: Understand the formula for circumference

The formula to calculate the circumference \( C \) of a circle is given by \( C = 2\pi r \), where \( r \) is the radius of the circle. In this problem, we know the radius \( r = 59.13 \) meters.

Step 2: Substitute the given values

Substitute the given radius and the approximation for \( \pi \) into the formula. This gives: \( C = 2 \times 3.14 \times 59.13 \).

Step 3: Perform the multiplication

Calculate the product \( 2 \times 3.14 \times 59.13 \). This simplifies to \( C = 370.2724 \).

Step 4: Round to the nearest hundredth

Round \( 370.2724 \) to the nearest hundredth, which results in \( 370.27 \).

Step 5: Interpret the result

The circumference of the crop circle is approximately \( 370.27 \) meters when rounded to the nearest hundredth.

Key Concepts

Circumference of a Circle

The circumference of a circle is like the perimeter of a circle - it is the total distance around the circle. This measurement is crucial in various fields, from geometry to practical applications like urban planning and agriculture. To find the circumference, we use the formula:\[ C = 2\pi r \]Here, \( C \) stands for the circumference, \( \pi \) is a constant approximately equal to 3.14, and \( r \) is the radius - the distance from the center of the circle to any point on its edge. This formula shows that the circumference is directly proportional to the radius. That means the larger the radius, the larger the circumference. Circle measurements are fundamental in problem-solving situations, especially when dealing with concepts like area and volume.

Using Pi in Calculations

Pi is a special number in mathematics. It represents the ratio of a circle's circumference to its diameter. This constant is approximately 3.14, but it is often better to use the symbol \( \pi \) in calculations for better precision. For practical exercises and simple calculations, we usually use 3.14.
When using \( \pi \) in real-world problems, like finding the circumference of a circle, it's important to choose whether to use 3.14 or a more exact version like 3.14159. We used 3.14 here for ease and typical classroom calculations. This approach simplifies the calculation and reduces the chances for error, especially when rounding is involved.

Radius and Diameter Calculations

In circle geometry, the radius and diameter are foundational concepts. Knowing one of these helps to easily calculate the other, as well as the circumference and area. The radius \( r \) is the distance from the center to the edge of the circle. In contrast, the diameter \( d \) is the distance across the circle, passing through the center, and is always twice the radius:\[ d = 2r \]Thus, if you know the radius, you simply double it to find the diameter. This information is critical as it allows you to switch between the two measurements based on what information you have or what is needed in a problem. Furthermore, knowing these values is essential for using the circumference formula \( C = 2\pi r \) or its equivalent using the diameter \( C = \pi d \). Understanding and mastering this relationship is key for solving geometric problems involving circles.