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 1,239,446.81 square feet.

Step 1: Determine if Diameter or Radius

The problem provides the diameter of the circle, which is 1256 ft. We need the radius to find the area.

Step 2: Calculate the Radius

The radius is half of the diameter. Therefore, the radius is calculated as follows: \[ r = \frac{d}{2} = \frac{1256}{2} = 628 \text{ ft} \]

Step 3: Use the Area Formula

To find the area of a circle, use the formula: \[ A = \pi r^2 \].With \( r = 628 \text{ ft} \), we substitute into the formula to get:\[ A = \pi (628)^2 \].

Step 4: Calculate the Area

Perform the calculations using the value of \( pi \) (approximately 3.14159):\[ A = 3.14159 \times (628)^2 \]\[ A \approx 3.14159 \times 394,384 \]\[ A \approx 1,239,446.8096 \text{ square feet} \]

Key Concepts

Understanding Diameter

In the world of circles, the diameter is a crucial measurement. It refers to the longest distance across the circle, passing directly through the center. Think of it as slicing the circle down its middle, from one edge to the opposite edge. This length is called the diameter, and it is usually denoted by the letter "d."

Here's why diameter is important:

• It helps in easily understanding the size of a circle.
• Knowing the diameter allows us to find the radius, which is essential for many calculations involving circles.

In most exercises, you might be given the diameter, and you can always find the radius from it. To do so, simply remember that the diameter is twice the radius. This leads us perfectly into the next concept—the radius.

Exploring the Radius

The radius is another fundamental feature of a circle, and it signifies half the length of the diameter. The radius extends from the center of the circle to any point on its circumference. It's essentially a building block for numerous circle-related formulas.

Given the diameter, calculating the radius is quite straightforward. You can use the simple formula: \[ r = \frac{d}{2} \] where "r" stands for the radius and "d" represents the diameter.

Knowing the radius is vital for many calculations, especially when finding areas, circumferences, or even working with sectors of circles. By understanding the radius, you unlock the potential to explore the circle more fully.

The Circle Area Formula

Once you have the radius at hand, determining the area of a circle is simple with the help of the circle area formula: \[ A = \pi r^2 \] where:

• "A" denotes the area.
• "r" refers to the radius.
• "\pi" (pi) is a constant, approximately equal to 3.14159.

This formula is derived from the relationship between the radius and the total space enclosed within the circle. By squaring the radius and multiplying it by \( \pi \), you calculate the number of square units needed to cover the circle's interior.

Thus, the concept of area using this formula shows how closely linked the radius and area of a circle are. With just the radius and the universal constant pi, you can determine how large or small any circle is.