Find the area of the sector of a circle of radius 5 cm, if the corresponding arc length is `3.5` cm.

Options

A. `7` `cm^2`
B. `8.75` `cm^2`
C. `8.25` `cm^2`
D. `7.25` `cm^2`

Explanation

To find the area of the sector of a circle with a radius of 5 cm and an arc length of 3.5 cm, we can follow these steps:

### Step 1: Understand the relationship between arc length, radius, and angle
The formula for the arc length \( L \) of a sector is given by:
\[ L = r \theta \]
where \( r \) is the radius and \( \theta \) is the angle in radians.

### Step 2: Substitute the known values to find \( \theta \)
Given:
- Radius \( r = 5 \) cm
- Arc length \( L = 3.5 \) cm

We can rearrange the formula to solve for \( \theta \):
\[ \theta = \frac{L}{r} = \frac{3.5}{5} \]
Calculating this gives:
\[ \theta = 0.7 \text{ radians} \]

### Step 3: Convert \( \theta \) from radians to degrees
To convert radians to degrees, we use the conversion factor \( \frac{180}{\pi} \):
\[ \theta = 0.7 \times \frac{180}{\pi} \]
Calculating this gives:
\[ \theta \approx 0.7 \times 57.2958 \approx 40.1 \text{ degrees} \]

### Step 4: Use the formula for the area of the sector
The area \( A \) of a sector is given by the formula:
\[ A = \frac{\theta}{360} \times \pi r^2 \]
Substituting the values we have:
\[ A = \frac{0.7 \times \frac{180}{\pi}}{360} \times \pi \times (5^2) \]
This simplifies to:
\[ A = \frac{0.7 \times 180 \times 25}{360} \]

### Step 5: Simplify the expression
Calculating the area:
\[ A = \frac{0.7 \times 180 \times 25}{360} = \frac{0.7 \times 25}{2} = \frac{17.5}{2} = 8.75 \text{ cm}^2 \]

### Final Answer
The area of the sector is approximately:
\[ \boxed{8.75 \text{ cm}^2} \]

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