To solve this problem, we will follow a systematic approach:
1. Calculate the Radian Measure of the Central Angle:
- The formula to find the radian measure ([tex]\(\theta\)[/tex]) of a central angle when you know the arc length ([tex]\(l\)[/tex]) and the radius ([tex]\(r\)[/tex]) is:
[tex]\[
\theta = \frac{l}{r}
\][/tex]
- Here, the arc length ([tex]\(l\)[/tex]) is 4 feet, and the radius ([tex]\(r\)[/tex]) is 7 feet. Therefore,
[tex]\[
\theta = \frac{4}{7} \approx 0.57 \text{ radians}
\][/tex]
2. Calculate the Area of the Sector:
- The formula to find the area of a sector ([tex]\(A\)[/tex]) when you know the radius ([tex]\(r\)[/tex]) and the radian measure of the central angle ([tex]\(\theta\)[/tex]) is:
[tex]\[
A = 0.5 \times r^2 \times \theta
\][/tex]
- Substituting the values, we get:
[tex]\[
A = 0.5 \times 7^2 \times 0.57
\][/tex]
- First, calculate [tex]\(7^2\)[/tex]:
[tex]\[
7^2 = 49
\][/tex]
- Next, calculate the product:
[tex]\[
0.5 \times 49 \times 0.57 = 0.5 \times 27.93 = 13.965
\][/tex]
- Rounding 13.965 to the nearest hundredth, we get:
[tex]\[
13.97 \text{ square feet}
\][/tex]
Therefore, the radian measure of the central angle subtended by the arc is [tex]\(\boxed{0.57}\)[/tex]. The area of the sector formed by the arc is [tex]\(\boxed{13.97}\)[/tex] square feet.
