Problem 48 - Elementary Algebra

Question

Set up an algebraic equation and then solve. The circumference of a circle measures 20 centimeters. Find the diameter rounded off to the nearest hundredth.

Step-by-Step Solution

Answer

The diameter is approximately 6.37 cm.

Step 1: Understand the formula for circumference

The formula for the circumference of a circle is given by \( C = \pi d \), where \( C \) is the circumference and \( d \) is the diameter of the circle.

Step 2: Substitute the circumference value

We are provided with the circumference \( C = 20 \) cm. Substitute this into the formula \( C = \pi d \) to get \( 20 = \pi d \).

Step 3: Solve for the diameter \( d \)

To find \( d \), rearrange the equation \( 20 = \pi d \) to solve for \( d \). This gives \( d = \frac{20}{\pi} \).

Step 4: Calculate the diameter

Use a calculator or approximation, \( \pi \approx 3.14159 \), to compute \( d = \frac{20}{3.14159} \approx 6.36619 \).

Step 5: Round the diameter

Round the computed value of the diameter to the nearest hundredth. This gives \( d \approx 6.37 \) cm.

Key Concepts

Circumference of a Circle

The circumference of a circle is the total distance around it. Imagine wrapping a string around the edge of a circle; the length of that string is the circumference. This concept is crucial because it relates directly to the size of the circle, comparable to how perimeter works for polygons. Knowing the circumference helps us understand other aspects like area and diameter. In mathematical terms, the formula for calculating the circumference is \( C = \pi d \), where \( C \) stands for circumference, \( \pi \) is a constant value, and \( d \) is the diameter of the circle. This formula allows us to quickly find how large a circle is as long as one of these values is known.

Diameter

The diameter is a key line that cuts the circle exactly in half, passing through its center. It is essentially the longest distance that can be measured across the circle. Understanding diameter is vital because it is a direct measure of the width of the circle. It connects closely with the circumference, as it's part of the formula \( C = \pi d \). So, if you have the circumference, you can find the diameter, and vice versa. It's also worth noting that the diameter is twice the radius (the distance from the center to any point on the edge of the circle), expressed as \( d = 2r \).

Pi (π)

Pi, denoted as \( \pi \), is an important mathematical constant. It's approximately equal to 3.14159, but it actually goes on infinitely without repeating. Pi is essential in the formula for the circumference \( C = \pi d \) and is fundamental when working with circles. The beauty of \( \pi \) lies in its universal application; it remains constant no matter the size of the circle. Its use allows us to transition between circumference and diameter, providing a bridge to solve various problems involving circles.

Problem-Solving Steps

Solving problems using algebraic equations involves a systematic approach. First, it's crucial to understand the problem and determine the information given. For example, if the circumference \( C = 20 \) cm is provided, this becomes your starting point. Next, substitute known values into the appropriate formula. Following this, rearrange the formula to isolate the unknown variable—in this case, the diameter \( d \). Calculate using the value of Pi to get a numerical answer. Be sure to round the final result appropriately, as specified in the problem. This structured method is essential for effectively solving equations and finding unknown values.