Problem 63 - Precalculus Mathematics for Calculus

Question

Find the area of a sector with central angle 1 rad in a circle of radius \(10 \mathrm{m}\)

Step-by-Step Solution

Answer

The area of the sector is 50 square meters.

Step 1: Understand the Sector Area Formula

The area of a sector with central angle \( \theta \) in a circle of radius \( r \) is given by the formula: \[ A = \frac{1}{2} r^2 \theta \] where \( \theta \) is in radians.

Step 2: Identify Given Variables

We are given that the radius \( r = 10 \) meters and the central angle \( \theta = 1 \) radian.

Step 3: Substitute Values into the Formula

Substitute \( r = 10 \) and \( \theta = 1 \) into the sector area formula: \[ A = \frac{1}{2} (10)^2 (1) \]

Step 4: Calculate the Area

Calculate the expression: \[ A = \frac{1}{2} \times 100 \times 1 = 50 \] Thus, the area of the sector is \(50\) square meters.

Key Concepts

Understanding Circle Geometry

Circle geometry involves exploring the unique attributes and measurements of a circle. A circle is a round, two-dimensional shape defined by all points that are equal distance from a center point. Several elements define circle geometry:

• Radius (r): The distance from the center of the circle to any point on its edge.
• Diameter: Twice the radius, stretching from one side of the circle to the other passing through the center.
• Circumference: The complete distance around the circle.
• Area: The space contained within the circle's boundaries.

In the context of a sector, which is essentially a 'slice' of the circle, these measurements become crucial. Each sector has its own central angle, and understanding these components can help you effectively calculate various properties of a sector, including its area.

The Concept of Radian Measure

Radian measure is a method of measuring angles based on the arc length they subtend at the circle's center. Unlike degrees, where a full circle is divided into 360 equal parts, radians relate directly to the properties of a circle. A circle has a total angle measurement of \( 2\pi \) radians. Radians are convenient in mathematics, particularly in calculus and trigonometry, because they simplify many formulas and calculations.

• Definition: One radian is the angle made at the center of a circle by an arc whose length is equal to the radius of the circle.
• Conversion: Degrees can be converted into radians using the formula: \( \theta_{rad} = \theta_{deg} \times \frac{\pi}{180} \).
• To calculate sector areas or lengths using radians, having a firm grasp of how radian measure works is vital.

Sector Area Calculation

The calculation of a sector's area requires knowing the circle's radius and its central angle, measured in radians. The formula for finding the area of a sector is straightforward and given by: \[ A = \frac{1}{2} r^2 \theta \] where \( A \) is the area, \( r \) is the radius, and \( \theta \) is the central angle in radians. Here's a quick breakdown of the process:

• Identify Variables: Start by identifying the radius and the central angle. These are typically provided or calculable from the problem.
• Plug into Formula: Substitute the known values into the sector area formula.
• Solve: Simplify to find the answer, ensuring that the central angle is in radians.

In the example from the exercise, with a radius of \(10 \mathrm{m}\) and a central angle of \(1\) radian, you substitute and solve to find that the area is \(50\) square meters. This approach highlights how radians simplify calculations in circular measurements.

Precalculus Context

Precalculus serves as the foundation for calculus, introducing concepts that become pivotal in higher-level mathematics. It bundles topics like algebra, geometry, and trigonometry to prepare students for the complexities of calculus. Understanding the sector area fits into precalculus because it combines algebraic manipulation with geometric principles.

• Foundational Skills: Precalculus solidifies skills in manipulating formulas and understanding geometric properties.
• Connection to Calculus: In calculus, the relationships and formulas from precalculus, like those for circle geometry, become tools for dealing with derivatives and integrals.
• Application and Problem-Solving: Precalculus hones your ability to apply mathematical concepts to solve problems efficiently, such as calculating areas using radian measure.

Building a strong understanding of these precalculus concepts ensures a smoother transition to more advanced topics, where intuition and familiarity with topics such as circle geometry and radians are key to success.