Sure, let's go through the steps to find the angle of the sector given the radius and the perimeter.
### Step 1: Understand the Perimeter Formulation
The perimeter \(\( P \)\) of a sector of a circle is given by:
\(\[ P = 2 \cdot \text{radius} + \text{arc length} \]\)
Now, the arc length can be determined from the angle \(\( \theta \)\) (in degrees) and the radius \(\( r \)\). The formula to calculate the arc length \(\( L \)\) is:
\(\[ L = \frac{\theta}{360} \cdot 2 \pi r \]\)
### Step 2: Express Perimeter in Terms of Angle \(\( \theta \)\)
Plug the arc length formula into the perimeter formula:
\(\[ P = 2r + \frac{\theta}{360} \cdot 2 \pi r \]\)
Here, the given values are:
- Radius \(\( r = 8 \)\) cm
- Perimeter \(\( P = 26 \)\) cm
### Step 3: Substitute the Known Values and Solve for \(\( \theta \)\)
Substitute \(\( r \)\) and \(\( P \)\) into the equation:
\(\[ 26 = 2 \cdot 8 + \frac{\theta}{360} \cdot 2 \pi \cdot 8 \]\)
\(\[ 26 = 16 + \frac{\theta}{360} \cdot 16 \pi \]\)
### Step 4: Simplify and Isolate \(\( \theta \)\)
First, simplify the equation:
\(\[ 26 = 16 + \frac{16 \pi \theta}{360} \]\)
Subtract 16 from both sides to isolate the term with \(\( \theta \)\):
\(\[ 10 = \frac{16 \pi \theta}{360} \]\)
To solve for \(\( \theta \)\), multiply both sides by 360 to get rid of the denominator:
\(\[ 3600 = 16 \pi \theta \]\)
Finally, divide both sides by \(\( 16 \pi \)\) to isolate \(\( \theta \)\):
\(\[ \theta = \frac{3600}{16 \pi} \]\)
### Step 5: Calculate the Value of \(\( \theta \)\)
When this calculation is performed, the angle \(\( \theta \)\) is:
\(\[ \theta \approx 71.62 \]\)
Thus, the angle of the sector is approximately \(\( 71.62 \)\) degrees.
