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