Problem 74 - Elementary Algebra

Question

For each of the following problems, use \(3.14\) as an approximation for \(\pi\). Your calculator should be of some help with these problems. Find the area of a circular plot of ground that has a radius \(16.3\) meters long. Express your answer to the nearest tenth of a square meter.

Step-by-Step Solution

Answer

The area is approximately 833.2 square meters.

Step 1: Understand the formula for area of a circle

The formula to find the area, \( A \), of a circle is given by \( A = \pi r^2 \), where \( r \) is the radius of the circle. Here, we will use \( 3.14 \) as an approximation for \( \pi \).

Step 2: Substitute the radius into the formula

Substitute \( r = 16.3 \) meters into the formula, which gives you \( A = 3.14 \times (16.3)^2 \).

Step 3: Calculate the square of the radius

First, calculate \( 16.3^2 \). This is \( 16.3 \times 16.3 = 265.69 \).

Step 4: Multiply by \( 3.14 \) to find the area

Next, multiply \( 265.69 \times 3.14 \) to find the area. This results in \( 833.2466 \).

Step 5: Round the Area to the Nearest Tenth

Finally, round \( 833.2466 \) to the nearest tenth, which gives \( 833.2 \).

Key Concepts

Circle Geometry

Understanding circles is a fundamental part of geometry. A circle is a simple shape, but it has several important characteristics:

• Center: The central point from which all points on the circle are equidistant.
• Radius: The distance from the center to any point on the circle's circumference. It is a crucial measurement when calculating the area or circumference.
• Diameter: The distance across the circle, passing through the center. It is twice the length of the radius.
• Circumference: The distance around the circle, often calculated as \(2\pi r\) or \(\pi d\).

When calculating the area of a circle, we primarily focus on the radius. The formula for the area, \(A = \pi r^2\), highlights how the circle's size grows with the square of its radius.

Mathematical Formulas

Formulas in mathematics are like recipes. They guide us in calculating different properties of geometric shapes. For a circle, two main formulas are often used:

• Area: The formula \(A = \pi r^2\) calculates how much space is inside the circle. This formula uses the radius to determine the square units within the circle.
• Circumference: The formula \(C = 2\pi r\) calculates the distance around the circle. It tells us how "long" the circle is if we were to "unroll" it into a straight line.

These formulas are integral to solving problems related to circles, allowing us to find exact values for area and circumference when we know the radius.

Approximations

In mathematics, especially when dealing with irrational numbers like \(\pi\), we often use approximations to make calculations manageable.
While \(\pi\) is approximately 3.14159, using 3.14 is a simple and efficient approximation for everyday calculations.

• Why Approximate: Approximations help simplify complex calculations and allow for easier computation, often without sacrificing significant accuracy for practical purposes.
• When to Use: For exercises, like calculating the area of a circle, using 3.14 for \(\pi\) is acceptable, especially when precision to several decimals isn't necessary.
• Impact of Approximation: Keep in mind that using 3.14 means the answer is slightly less precise than if using more digits of \(\pi\). This is usually fine for general purposes, such as the classroom setting.

By understanding when and why to approximate, students can effectively tackle practical problems without getting bogged down by overly complex numbers.