Circular Sector Arc Length Calculation A Step-by-Step Guide

by Scholario Team 60 views

Hey guys! Let's dive into a fascinating problem involving circular sectors, angles, and arc lengths. This is a classic math question that combines geometry and a bit of algebra, perfect for sharpening your problem-solving skills. We'll break down the problem step by step, making sure everyone can follow along. So, grab your thinking caps, and let's get started!

The Problem Unveiled

Let's kick things off by stating the problem clearly. We're dealing with a circular sector where the central angle measures x radians, and the radius is (x + 1) cm. Here's the twist the area of this sector is numerically equal to the circular measure of the central angle. Our mission, should we choose to accept it, is to figure out the length of the arc. We have a few options to choose from:

  • a) √2 cm
  • b) (√2 + 1) cm
  • c) (2-√2) cm
  • d) (√2 - 1) cm

Sounds intriguing, right? Now, let’s put on our detective hats and solve this mystery together!

Decoding the Fundamentals of Circular Sectors

Before we jump into solving the problem, let's quickly recap the basics of circular sectors. This will ensure we're all on the same page and have the necessary tools at our disposal. Think of a circular sector as a slice of pie cut from a circular pizza. It’s defined by two radii and the arc connecting their endpoints.

Key Elements of a Circular Sector

  • Central Angle (θ): This is the angle formed at the center of the circle between the two radii. It's usually measured in radians for mathematical calculations. Radians, guys, are a super important concept in trigonometry and calculus, so make sure you're comfy with them!
  • Radius (r): This is the distance from the center of the circle to any point on the circle's edge (including the endpoints of the arc).
  • Arc Length (s): This is the distance along the curved edge of the sector. It's like measuring the crust of our pizza slice.
  • Area (A): This is the space enclosed within the sector. It's the cheesy goodness of our pizza slice!

Essential Formulas to Remember

To tackle our problem, we need to know the formulas for arc length and the area of a circular sector. These are our secret weapons!

  1. Arc Length (s): The arc length is directly proportional to the central angle and the radius. The formula is:

    s = rθ

    Where:

    • s is the arc length
    • r is the radius
    • θ is the central angle in radians

  2. Area (A): The area of a circular sector is related to the central angle and the radius as well. The formula is:

    A = (1/2)r²θ

    Where:

    • A is the area
    • r is the radius
    • θ is the central angle in radians

With these formulas in our arsenal, we're ready to attack the problem head-on!

Cracking the Code: Solving the Problem Step-by-Step

Okay, let's get down to business and solve this problem. Remember, the key is to break it down into manageable steps and use the information we have wisely.

1. Identifying the Givens

First, let's clearly identify what the problem has given us:

  • Central angle (θ) = x radians
  • Radius (r) = (x + 1) cm
  • Area (A) = x (since the area is numerically equal to the central angle)

2. Applying the Area Formula

We know the area formula, and we know the values for A, r, and θ in terms of x. Let's plug them in:

  • A = (1/2)r²θ
  • x = (1/2)(x + 1)² x

3. Simplifying the Equation

Now comes the algebraic fun! Let's simplify the equation and solve for x. This might involve some expanding and rearranging.

  • x = (1/2)(x² + 2x + 1) x

Since x represents an angle in radians, and for a sector to exist, x must be greater than 0, we can divide both sides by x:

  • 1 = (1/2)(x² + 2x + 1)
  • 2 = x² + 2x + 1
  • 0 = x² + 2x - 1

4. Solving the Quadratic Equation

We've got a quadratic equation! Time to use the quadratic formula (or complete the square, if you're feeling fancy). Remember the quadratic formula? It's:

  • x = (-b ± √(b² - 4ac)) / 2a

In our equation, a = 1, b = 2, and c = -1. Plugging these values in:

  • x = (-2 ± √(2² - 4(1)(-1))) / 2(1)
  • x = (-2 ± √(8)) / 2
  • x = (-2 ± 2√2) / 2
  • x = -1 ± √2

5. Choosing the Valid Solution

We have two possible solutions for x: -1 + √2 and -1 - √2. However, since x represents an angle, it must be positive. Therefore, we choose:

  • x = -1 + √2

6. Calculating the Arc Length

We're almost there! Now that we have x, we can find the radius (r) and then use the arc length formula.

  • r = x + 1 = (-1 + √2) + 1 = √2 cm

Now, plug r and x into the arc length formula:

  • s = rθ
  • s = (√2)(-1 + √2)
  • s = -√2 + 2
  • s = 2 - √2 cm

7. The Grand Finale: Selecting the Correct Answer

Drumroll, please! Looking back at our options, the arc length we calculated (2 - √2) cm matches option c) (2-√2) cm. We did it!

Why This Problem Matters: Real-World Applications

Okay, so we solved a math problem – awesome! But you might be wondering,