The area of the bottom of the box (the shaded region) not covered by donuts will be 13.94 in²
Explanation
1. Across the length of the box, there are three filled donuts. That means the total diameter of three filled donuts = 9 inch
So, diameter of each filled donuts = [tex]\frac{9}{3} = 3[/tex] inch and the radius [tex]= \frac{3}{2}= 1.5[/tex] inch.
Area of each filled donuts = [tex]\pi r^2 = \pi (1.5)^2 = 2.25\pi[/tex] in²
So, the area covered by three filled donuts [tex]= 3*2.25\pi = 21.21[/tex] in²
2. Diameter of each glazed donuts = 3 in and diameter of each small circle inside = 1 in
So, the radius of each glazed donuts [tex]= \frac{3}{2}= 1.5[/tex] in and radius of each small circle [tex]= \frac{1}{2}= 0.5[/tex] in
So, the area of each glazed donuts [tex]= \pi (1.5)^2 - \pi (0.5)^2 = 2.25\pi -0.25\pi = 2\pi[/tex] in²
The Area covered by three glazed donuts [tex]= 3* 2\pi = 6\pi = 18.85[/tex] in²
3. Total area covered by 6 donuts = 21.21 + 18.85 = 40.06 in²
Area of the box = (length) × (width) = (9 × 6)in² = 54 in²
So, the area of the bottom of the box (the shaded region) not covered by donuts = (54 - 40.06) in² = 13.94 in²
