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Calculate the arc length [tex]\( S \)[/tex] and area [tex]\( A \)[/tex] of the following sector:

[tex]\[
\begin{array}{l}
\alpha = 0.7 \, \text{rad} \\
r = 170 \, \text{ft}
\end{array}
\][/tex]

[tex]\[
S = \square \, \text{ft}
\][/tex]
(Round to the nearest whole number as needed.)

[tex]\[
A = \square \, \text{sqft}
\][/tex]
(Round to the nearest whole number as needed.)



Answer :

GinnyAnswer
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]

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