Problem 94 - Precalculus

Question

Find the area of the circular sector given the indicated radius and central angle. Round your answers to three significant digits. $$\theta=14^{\circ}, r=3.0 \mathrm{ft}$$

Step-by-Step Solution

Answer

The area of the sector is approximately 1.10 square feet.

Step 1: Convert Degrees to Radians

To find the area of a sector, we first need the angle in radians. The formula to convert degrees to radians is: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \). For \( \theta = 14^\circ \), we have: \[ 14^\circ \times \frac{\pi}{180} = \frac{14\pi}{180} = \frac{\pi}{12.857} \approx 0.2443 \text{ radians} \]

Step 2: Use the Area of a Circular Sector Formula

The formula to calculate the area of a circular sector is: \( A = \frac{1}{2} r^2 \theta \), where \( r \) is the radius and \( \theta \) is the angle in radians. Substitute the values \( r = 3.0 \) ft and \( \theta \approx 0.2443 \): \[ A = \frac{1}{2} (3.0)^2 (0.2443) = \frac{1}{2} \times 9 \times 0.2443 = 1.0984 \text{ square feet} \]

Step 3: Round the Area to Three Significant Digits

Our final step is to round the calculated area to three significant digits. The area is approximately \( 1.0984 \) square feet, so rounding gives us \( 1.10 \text{ square feet} \).

Key Concepts

Angle Conversion to Radians

When dealing with circular sectors, angles are often presented in degrees. However, to properly use formulas in mathematics, especially those involving circles, it's crucial to work with radians. Radians are a unit of angular measure based on the radius of a circle. This conversion is essential because most mathematical formulas, like those involving trigonometric functions, require angles to be in radians.
To convert an angle from degrees to radians, you use the formula:

• \( ext{radians} = ext{degrees} \times \frac{\pi}{180} \)

This equation substitutes degrees with their equivalent in radians using \( \pi \), where \( \pi \approx 3.14159 \).
For instance, converting 14 degrees to radians involves:

• \( 14 \times \frac{\pi}{180} \approx 0.2443 \)

This conversion is key for further calculations involving sector areas, velocity and more, making it a fundamental step in trigonometric and calculus problems.

Circular Sector Formula

The area of a circular sector is part of the circle defined by its central angle and radius. This area can be found using the circular sector formula:

• \( A = \frac{1}{2} r^2 \theta \)

Here, \( A \) represents the area of the sector, \( r \) is the radius, and \( \theta \) is the central angle in radians. This formula essentially stems from the area of the full circle, \( \pi r^2 \), scaled by the fraction of the circle's angle that the sector represents.
For example, if the radius \( r \) is 3 feet and the angle \( \theta \) is approximately 0.2443 radians, you plug these into the formula:

• \( A = \frac{1}{2} \times 3^2 \times 0.2443 \approx 1.0984 \text{ square feet} \)

Understanding this concept allows you to calculate pieces of a circle new and fascinating nature-related measurements.

Significant Figures in Mathematics

Significant figures are a way to express precision in numerical calculations. They indicate how many digits are meaningful in a number and help in determining the accuracy of that measurement or calculation.
When rounding numbers, you typically follow certain rules. For the number 1.0984, rounding to three significant figures involves looking at the fourth figure, "8" in this case. Since it's 5 or more, you round up, resulting in 1.10.

• Start counting significant figures from the first non-zero digit.
• Include all digits up to the required number, rounding as necessary.

This practice ensures that your final answers are both accurate and precise, which is crucial in engineering, science, and real-world applications. Rounding to a specific number of significant figures also keeps the data's integrity while simplifying its complexity, making it easier to communicate and understand.