To determine the area of the remaining portion of a circle after half of it has been removed, we need to perform the following steps for each original circle area:
1. Identify the total area of the circle – This is provided in the problem statement for each individual circle.
2. Calculate the area of half of the circle – This is achieved by dividing the original area by 2.
3. Note that the shaded region is exactly half of the original circle area.
Let's apply these steps to each of the given circle areas:
### For an original area of 60 square feet:
1. Original area of the circle: [tex]\( 60 \text{ square feet} \)[/tex]
2. Area of half of the circle: [tex]\( \frac{60}{2} = 30 \text{ square feet} \)[/tex]
3. Therefore, the area of the remaining portion: [tex]\( 30 \text{ square feet} \)[/tex]
### For an original area of 120 square feet:
1. Original area of the circle: [tex]\( 120 \text{ square feet} \)[/tex]
2. Area of half of the circle: [tex]\( \frac{120}{2} = 60 \text{ square feet} \)[/tex]
3. Therefore, the area of the remaining portion: [tex]\( 60 \text{ square feet} \)[/tex]
### For an original area of 225 square feet:
1. Original area of the circle: [tex]\( 225 \text{ square feet} \)[/tex]
2. Area of half of the circle: [tex]\( \frac{225}{2} = 112.5 \text{ square feet} \)[/tex]
3. Therefore, the area of the remaining portion: [tex]\( 112.5 \text{ square feet} \)[/tex]
### For an original area of 450 square feet:
1. Original area of the circle: [tex]\( 450 \text{ square feet} \)[/tex]
2. Area of half of the circle: [tex]\( \frac{450}{2} = 225 \text{ square feet} \)[/tex]
3. Therefore, the area of the remaining portion: [tex]\( 225 \text{ square feet} \)[/tex]
### Summary
The approximate areas of the remaining portions of the circles are:
- For [tex]\( 60 \text{ square feet} \)[/tex]: [tex]\( 30 \text{ square feet} \)[/tex]
- For [tex]\( 120 \text{ square feet} \)[/tex]: [tex]\( 60 \text{ square feet} \)[/tex]
- For [tex]\( 225 \text{ square feet} \)[/tex]: [tex]\( 112.5 \text{ square feet} \)[/tex]
- For [tex]\( 450 \text{ square feet} \)[/tex]: [tex]\( 225 \text{ square feet} \)[/tex]
