Sure, let’s calculate the arc length [tex]\( S \)[/tex] and the area [tex]\( A \)[/tex] of the sector step-by-step.
### 1. Calculation of the Arc Length [tex]\( S \)[/tex]:
The formula to find the arc length [tex]\( S \)[/tex] of a sector is given by:
[tex]\[ S = \theta \cdot r \][/tex]
where:
- [tex]\( \theta \)[/tex] is the central angle in radians
- [tex]\( r \)[/tex] is the radius of the circle
Given:
[tex]\[
\theta = 0.7 \text{ rad}
\][/tex]
[tex]\[
r = 170 \text{ ft}
\][/tex]
Substituting these values into the formula:
[tex]\[
S = 0.7 \times 170
\][/tex]
When we calculate this, we get:
[tex]\[
S \approx 119 \text{ ft}
\][/tex]
### 2. Calculation of the Area [tex]\( A \)[/tex]:
The formula to find the area [tex]\( A \)[/tex] of a sector is given by:
[tex]\[ A = \frac{1}{2} r^2 \theta \][/tex]
where:
- [tex]\( r \)[/tex] is the radius of the circle
- [tex]\( \theta \)[/tex] is the central angle in radians
Given:
[tex]\[
r = 170 \text{ ft}
\][/tex]
[tex]\[
\theta = 0.7 \text{ rad}
\][/tex]
Substituting these values into the formula:
[tex]\[
A = \frac{1}{2} \times 170^2 \times 0.7
\][/tex]
First, compute [tex]\( 170^2 \)[/tex]:
[tex]\[
170^2 = 28900
\][/tex]
Then compute:
[tex]\[
A = \frac{1}{2} \times 28900 \times 0.7
\][/tex]
When we carry out this calculation, we get:
[tex]\[
A \approx 10115 \text{ sqft}
\][/tex]
### Final Answers:
[tex]\[
S \approx 119 \text{ ft}
\][/tex]
[tex]\[
A \approx 10115 \text{ sqft}
\][/tex]
Thus:
[tex]\[
S = 119 \text{ ft}
\][/tex]
[tex]\[
A = 10115 \text{ sqft}
\][/tex]
