Problem 120 - A Graphical Approach to Precalculus with Limits

Question

Approximate the area of a sector of a circle having radius \(r\) and central angle \(\boldsymbol{\theta}.\) \(r=59.8\) kilometers \(; \theta=\frac{2 \pi}{3}\) radians

Step-by-Step Solution

Answer

The approximate area of the sector is 3741.81 square kilometers.

Step 1: Understand the formula for the area of a sector

The area of a sector of a circle is calculated using the formula \( A = \frac{1}{2} r^2 \theta \), where \( r \) is the radius and \( \theta \) is the central angle in radians.

Step 2: Substitute the given values into the formula

Using the formula from Step 1, substitute \( r = 59.8 \) and \( \theta = \frac{2\pi}{3} \) into the formula. This gives: \[ A = \frac{1}{2} \times (59.8)^2 \times \frac{2\pi}{3} \].

Step 3: Calculate the area of the sector

First, calculate \( (59.8)^2 = 3576.04 \). Then, multiply it by \( \frac{2\pi}{3} \) and then by \( \frac{1}{2} \). Therefore: \[ A = \frac{1}{2} \times 3576.04 \times \frac{2\pi}{3} \approx 3741.81\].

Key Concepts

Radius of a Circle

The radius of a circle is a fundamental concept in geometry. It is the distance from the center of the circle to any point on its circumference. An understanding of the radius is crucial when grasping circle-based geometrical concepts, such as the area of a sector.
In more intuitive terms, think of the radius as a line that stretches from the very center to the boundary of a circle, almost like the spoke of a wheel.

• The radius is usually denoted by the letter "r".
• It determines the size of the circle. A larger radius means a larger circle.
• In the formula to calculate the area of a sector, the radius is squared. This is because we are interested in a 2-dimensional area.

Whenever you're dealing with problems involving circles, identifying the radius is often your first step in applying necessary formulas.

Central Angle in Radians

The central angle of a circle is another key component in geometric calculations, particularly when finding areas such as that of a sector. A central angle is formed by two radii extending from the center of the circle and cuts off an arc along the circumference. It is typically measured in radians.
Radians provide a natural way of measuring angles as they relate directly to the length of the arc they intercept, with one full circle equal to approximately 6.2832 radians (or 2π radians).

• Radians arise from dividing the length of the arc by the radius of the circle - hence it's a ratio without units.
• Since radians are dimensionless, they fit neatly into various formulas in geometry and calculus.
• In one circle, there are 2π radians, making them easier and often more logical for calculations involving circular arcs or sectors.

Understanding how to work with radians and converting between degrees and radians, if necessary, provides a firmer grip on solving geometric problems effectively.

Geometry Formulas

Geometry formulas are crucial tools in solving mathematical problems related to shapes and spaces. When it comes to circular forms, such as finding the area of a sector, specific geometry formulas become essential. These formulas make it possible to translate a circle's dimensions into meaningful numerical values.
One frequently used formula in this context is the area of a sector formula: \[ A = \frac{1}{2} r^2 \theta \] where \(A\) is the area of the sector, \(r\) is the radius, and \(\theta\) is the central angle in radians.

• This formula relies on both the concept of radius and the radian measure of the angle to provide the sector's proportion of the entire circle's area.
• It assumes that the circle’s whole area is \(\pi r^2\), making the sector's area a fraction depending on \(\theta\).
• Applying the formula involves substituting the known values for \(r\) and \(\theta\) to find \(A\).

Mastery of these geometry formulas, like the one for sector area, is beneficial. They are not used in isolation but as part of broader problem-solving strategies in geometry and calculus.