https://file.aiquickdraw.com/mathbot/file/17288376568332s85wrvn.jpg

Let's break down and solve the problem step by step based on the image you've provided.

### Part 1: Radian Measure of the Central Angle
The formula for the length of an arc (\(s\)) is:
\[
s = r \theta
\]
where:
- \(s\) is the arc length (given as 4 feet),
- \(r\) is the radius of the circle (given as 7 feet),
- \(\theta\) is the central angle in radians (what we need to find).

Rearranging the formula to solve for \(\theta\):
\[
\theta = \frac{s}{r} = \frac{4}{7} \text{ radians}.
\]
So, the radian measure of the central angle is approximately:
\[
\theta \approx 0.5714 \text{ radians}.
\]

### Part 2: Area of the Sector
The formula for the area of a sector is:
\[
A = \frac{1}{2} r^2 \theta
\]
Substitute the values:
- \(r = 7\) feet,
- \(\theta = 0.5714\) radians.

Thus:
\[
A = \frac{1}{2} \times 7^2 \times 0.5714 = \frac{1}{2} \times 49 \times 0.5714 = 24.5714 \text{ square feet}.
\]
So, the area of the sector is approximately 24.57 square feet when rounded to the nearest hundredth.

### Final Answers:
1. **Radian measure of the central angle:** 0.57 radians.
2. **Area of the sector:** 24.57 square feet.

Would you like more details or have any questions?

Here are 5 related questions to further your understanding:
1. How would the radian measure change if the arc length was doubled?
2. What would the area of the sector be if the radius was increased to 10 feet?
3. How would you find the central angle in degrees instead of radians?
4. What is the formula for the circumference of a circle, and how does it relate to this problem?
5. How does the sector's area compare to the area of the whole circle?

**Tip:** Remember that for small angles, radians give a more natural measure for circular calculations than degrees because they directly relate the arc length to the radius.

In a circle with a radius of 7 feet, the radian measure of the central angle subtended by an arc with a length of 4 feet is ______ radians. The area of the sector formed by the arc is ______ square feet. Assume π = 3.14, and round your answers to the nearest hundredth.

Solve a problem involving a circle with a radius of 7 feet, where an arc length of 4 feet subtends a central angle in radians. Learn how to calculate the radian measure and the area of the sector using key geometry formulas. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation