Problem 73 - Precalculus with Calculus Previews

Question

In Problems \(73-76\), find the area of the circular sector having the given radius \(r\) and central angle \(\theta\). $$ r=3 \mathrm{ft}, \theta=7.2 \text { radians } $$

Step-by-Step Solution

Answer

The area of the circular sector is 32.4 square feet.

Step 1: Understanding the Formula

The area of a circular sector can be 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

Substitute the given values of \( r = 3 \) ft and \( \theta = 7.2 \) radians into the formula. This gives us \( A = \frac{1}{2} \times (3)^2 \times 7.2 \).

Step 3: Calculate the Area

First, square the radius: \( 3^2 = 9 \). Then, plug this into our equation to get \( A = \frac{1}{2} \times 9 \times 7.2 \). Finally, calculate the product: \( A = \frac{1}{2} \times 64.8 = 32.4 \).

Step 4: Write the Final Answer

The area of the circular sector is 32.4 square feet.

Key Concepts

Radius

The radius is a crucial element when calculating the area of a circular sector. In geometry, the radius of a circle is the distance from the center of the circle to any point on its circumference. It is often denoted by the letter \( r \). The radius is important because it helps to determine both the size of the circle and the size of any sector within that circle. For example, if you have a circle with a larger radius, the circle itself is bigger, as is any sector that is a part of it.

In our exercise, the radius \( r \) is given as 3 feet. This means that from the center of the circle to the edge, the distance is exactly 3 feet. When calculating the area of the sector, squaring the radius is part of the formula: \( r^2 \). Squaring the radius in this case gives us \( 3^2 = 9 \), which is then used in further calculations to find the area.

Central Angle in Radians

Understanding the central angle in radians is essential to finding the area of a circular sector. The central angle is the angle that is subtended by two radii at the center of the circle, and when expressed in radians, it gives a direct measure of the arc it subtends, based on the circle's radius.

Radian is a measure of angle based on the radius of the circle, where one radian is the angle created when the arc length is equal to the radius. Therefore, radians provide a natural way to express angles with respect to the circle itself.

• One full circle is \( 2\pi \) radians, which equals 360 degrees.
• Thus, smaller divisions like 7.2 radians indicate a portion of the circle's circumference.

In our exercise, the central angle is specified as \( \theta = 7.2 \) radians, which is used directly in the area calculation. The formula \( A = \frac{1}{2} r^2 \theta \) shows how this angle in radians directly contributes to determining the sector area.

Area Formula for a Circular Sector

The area formula for a circular sector is a key component of solving these types of mathematical problems. It is given by the formula \( A = \frac{1}{2} r^2 \theta \), where \( A \) represents the area of the sector, \( r \) is the radius of the circle, and \( \theta \) is the central angle in radians.

This formula allows us to calculate only the portion of the circle defined by the central angle. Here is how the formula works step by step:

• The term \( r^2 \) involves squaring the radius to adjust for the unit areas of the circle's size.
• \( \theta \) in radians helps determine how much of the circle's circumference the sector covers.
• The factor \( \frac{1}{2} \) adjusts the calculation because a sector is essentially a fraction of the full circle.

By substituting the values \( r = 3 \) feet and \( \theta = 7.2 \) radians into the formula, we find the area to be \( 32.4 \) square feet, indicating the exact size of this portion of the circle.