Problem 16 - Basic Technical Mathematics with Calculus

Question

Find the area of the circle with the given radius or diameter. $$d=1256 \mathrm{ft}$$

Step-by-Step Solution

Answer

The area of the circle is approximately 1,238,705.69 square feet.

Step 1: Understand the Relationship

The diameter of a circle is twice the length of its radius. If the diameter is given by \(d\), then the radius \(r\) can be calculated as \(r = \frac{d}{2}\).

Step 2: Calculate the Radius

Given the diameter \(d = 1256 \text{ ft}\), the radius \(r\) will be \(r = \frac{1256}{2} = 628 \text{ ft}\).

Step 3: Use Area Formula for Circle

The formula to find the area of a circle is \(A = \pi r^2\), where \(r\) is the radius.

Step 4: Substitute the Radius

Substitute the radius into the area formula: \(A = \pi (628)^2\).

Step 5: Calculate and Simplify

First, calculate \( (628)^2 = 394,384 \). Then, multiply by \( \pi \) to get \( A = \pi \times 394,384 \approx 1,238,705.69 \text{ ft}^2\) using \( \pi \approx 3.14159 \).

Key Concepts

Radius and Diameter Relationship

In geometry, understanding the relationship between the radius and diameter of a circle is essential. When dealing with circles, the radius is defined as the distance from the center of the circle to any point on its boundary. The diameter is simply twice the length of the radius. Hence, if you know one, you can easily find the other.

To find the radius when the diameter is given, you can use the formula:

• Radius ( \( r \) ) = Diameter ( \( d \) ) / 2

This formula allows you to transform diameter measurements into radius figures easily, which is crucial when calculating a circle's area. For example, if a circle has a diameter of 1256 ft, the radius would be 628 ft. This basic understanding forms a foundation for solving more complex geometry problems involving circles.

Always remember, knowing either the radius or diameter gives you complete control over circle calculations!

Area Formula for Circles

Once you have the radius of a circle, finding its area becomes a straightforward process. The area of a circle is a measure of the space contained within its boundary. It is calculated using a specific formula:

• \( A = \pi r^2 \)

In this formula, \( A \) represents the area, \( \pi \) is a constant approximately equal to 3.14159, and \( r \) is the radius of the circle.

To find the area, follow these steps:- Square the radius to get the term \( r^2 \).- Multiply \( r^2 \) by \( \pi \).This will yield the area in square units, matching the units of the radius. For a circle with a radius of 628 ft, you would first calculate \( 628^2 = 394,384 \), and then compute the area as \( A = \pi \times 394,384 \approx 1,238,705.69 \text{ ft}^2 \).

Understanding and using this formula is vital in any problem involving circular measurements, making complex area evaluations manageable.

Geometry Problem Solving

Solving geometry problems involving circles requires both an understanding of formulas and a systematic approach. Let's break down the process using our example of finding a circle's area provided its diameter.

First, identify what's given and what's needed. In this instance, we're given the diameter and need the area.

• Convert the diameter to radius using the formula \( r = \frac{d}{2} \).
• Apply the area formula \( A = \pi r^2 \) using the radius.
  
  The key to solving any geometry problem is to move step by step, ensuring that each part of the formula is correctly applied. For example, with a diameter of 1256 ft, you properly convert it to a radius of 628 ft, then calculate the area using \( 628^2 \times \pi \). Use approximations for better precision, such as \( \pi \approx 3.14159 \), to finalize the result as \( 1,238,705.69 ft^2 \).

This structured approach simplifies complex problems and ensures accurate solutions, especially when dealing with large numbers or exact measurements in geometry. Remember, clarity and order can lead to mastering circle-related problems!