Answer :
To solve the problem of finding the area of Pacman's body when his mouth is open at 70 degrees, we need to follow a series of steps. First, we'll calculate the area of the entire circle representing Pacman's body if his mouth were closed. Then we will find the area of the sector corresponding to his open mouth. Finally, we subtract the area of the mouth from the total area of the circle to get the remaining area of Pacman's body. Here is the step-by-step solution:
### Step 1: Calculate the Area of the Entire Circle
The area [tex]\( A \)[/tex] of a circle can be found using the formula:
[tex]\[ A = \pi r^2 \][/tex]
where [tex]\( r \)[/tex] is the radius of the circle.
Here, the radius [tex]\( r \)[/tex] is given as 8 mm.
[tex]\[ A = \pi \times 8^2 \][/tex]
[tex]\[ A = \pi \times 64 \][/tex]
[tex]\[ A = 64\pi \][/tex]
### Step 2: Convert the Central Angle from Degrees to Radians
The central angle of Pacman's mouth is given as 70 degrees. We need to convert this angle to radians to use it in our sector area calculation. We use the conversion factor:
[tex]\[ \text{Radians} = \text{Degrees} \times \left( \frac{\pi}{180} \right) \][/tex]
So,
[tex]\[ \text{Angle in radians} = 70 \times \left( \frac{\pi}{180} \right) \][/tex]
[tex]\[ \text{Angle in radians} = \frac{70\pi}{180} \][/tex]
[tex]\[ \text{Angle in radians} = \frac{7\pi}{18} \][/tex]
### Step 3: Calculate the Area of the Sector Representing the Mouth
The area [tex]\( A_{\text{sector}} \)[/tex] of a sector of a circle can be found using the formula:
[tex]\[ A_{\text{sector}} = \left( \frac{\theta}{2\pi} \right) \times \pi r^2 \][/tex]
where [tex]\( \theta \)[/tex] is the central angle in radians.
Here, [tex]\( \theta = \frac{7\pi}{18} \)[/tex] and [tex]\( r = 8 \)[/tex] mm.
[tex]\[ A_{\text{sector}} = \left( \frac{\frac{7\pi}{18}}{2\pi} \right) \times \pi \times 64 \][/tex]
[tex]\[ A_{\text{sector}} = \left( \frac{7\pi}{36\pi} \right) \times 64\pi \][/tex]
[tex]\[ A_{\text{sector}} = \left( \frac{7}{36} \right) \times 64\pi \][/tex]
[tex]\[ A_{\text{sector}} = \frac{448\pi}{36} \][/tex]
[tex]\[ A_{\text{sector}} = \frac{56\pi}{9} \][/tex]
### Step 4: Calculate the Remaining Area of Pacman's Body
Finally, we subtract the area of the mouth (sector) from the total area of the circle to find the remaining area of Pacman's body.
[tex]\[ A_{\text{body}} = 64\pi - \frac{56\pi}{9} \][/tex]
Let's find a common denominator to subtract these areas:
[tex]\[ A_{\text{body}} = \frac{576\pi}{9} - \frac{56\pi}{9} \][/tex]
[tex]\[ A_{\text{body}} = \frac{520\pi}{9} \][/tex]
Thus, the area of Pacman's body is:
[tex]\[ \boxed{\frac{520\pi}{9} \text{ mm}^2} \][/tex]
This completes the step-by-step solution of finding the area of Pacman's body when his mouth is open at a 70° angle.
### Step 1: Calculate the Area of the Entire Circle
The area [tex]\( A \)[/tex] of a circle can be found using the formula:
[tex]\[ A = \pi r^2 \][/tex]
where [tex]\( r \)[/tex] is the radius of the circle.
Here, the radius [tex]\( r \)[/tex] is given as 8 mm.
[tex]\[ A = \pi \times 8^2 \][/tex]
[tex]\[ A = \pi \times 64 \][/tex]
[tex]\[ A = 64\pi \][/tex]
### Step 2: Convert the Central Angle from Degrees to Radians
The central angle of Pacman's mouth is given as 70 degrees. We need to convert this angle to radians to use it in our sector area calculation. We use the conversion factor:
[tex]\[ \text{Radians} = \text{Degrees} \times \left( \frac{\pi}{180} \right) \][/tex]
So,
[tex]\[ \text{Angle in radians} = 70 \times \left( \frac{\pi}{180} \right) \][/tex]
[tex]\[ \text{Angle in radians} = \frac{70\pi}{180} \][/tex]
[tex]\[ \text{Angle in radians} = \frac{7\pi}{18} \][/tex]
### Step 3: Calculate the Area of the Sector Representing the Mouth
The area [tex]\( A_{\text{sector}} \)[/tex] of a sector of a circle can be found using the formula:
[tex]\[ A_{\text{sector}} = \left( \frac{\theta}{2\pi} \right) \times \pi r^2 \][/tex]
where [tex]\( \theta \)[/tex] is the central angle in radians.
Here, [tex]\( \theta = \frac{7\pi}{18} \)[/tex] and [tex]\( r = 8 \)[/tex] mm.
[tex]\[ A_{\text{sector}} = \left( \frac{\frac{7\pi}{18}}{2\pi} \right) \times \pi \times 64 \][/tex]
[tex]\[ A_{\text{sector}} = \left( \frac{7\pi}{36\pi} \right) \times 64\pi \][/tex]
[tex]\[ A_{\text{sector}} = \left( \frac{7}{36} \right) \times 64\pi \][/tex]
[tex]\[ A_{\text{sector}} = \frac{448\pi}{36} \][/tex]
[tex]\[ A_{\text{sector}} = \frac{56\pi}{9} \][/tex]
### Step 4: Calculate the Remaining Area of Pacman's Body
Finally, we subtract the area of the mouth (sector) from the total area of the circle to find the remaining area of Pacman's body.
[tex]\[ A_{\text{body}} = 64\pi - \frac{56\pi}{9} \][/tex]
Let's find a common denominator to subtract these areas:
[tex]\[ A_{\text{body}} = \frac{576\pi}{9} - \frac{56\pi}{9} \][/tex]
[tex]\[ A_{\text{body}} = \frac{520\pi}{9} \][/tex]
Thus, the area of Pacman's body is:
[tex]\[ \boxed{\frac{520\pi}{9} \text{ mm}^2} \][/tex]
This completes the step-by-step solution of finding the area of Pacman's body when his mouth is open at a 70° angle.
