Question
One rectangle is "framed" within another. Find the area of the shaded region if the "frame" is 3 units wide.
the frame is 12 x 10
THE ANSWER ISNT 96 IVE TRIED AND IT DNT WORK
the frame is 12 x 10
THE ANSWER ISNT 96 IVE TRIED AND IT DNT WORK
Answers
GPT-4o mini
To find the area of the shaded region, we can start by calculating the area of the larger rectangle and the area of the smaller, framed rectangle inside it.
1. **Area of the larger rectangle**:
The larger rectangle has dimensions 12 units by 10 units, so its area is:
\[
\text{Area}_{\text{large}} = \text{length} \times \text{width} = 12 \times 10 = 120 \, \text{square units}
\]
2. **Dimensions of the smaller rectangle**:
Since the frame is 3 units wide, we need to subtract 3 units from each side of the larger rectangle to find the dimensions of the smaller rectangle.
- The width of the smaller rectangle is:
\[
\text{Width}_{\text{small}} = \text{Width}_{\text{large}} - 2 \times \text{frame width} = 10 - 2 \times 3 = 10 - 6 = 4 \, \text{units}
\]
- The length of the smaller rectangle is:
\[
\text{Length}_{\text{small}} = \text{Length}_{\text{large}} - 2 \times \text{frame width} = 12 - 2 \times 3 = 12 - 6 = 6 \, \text{units}
\]
3. **Area of the smaller rectangle**:
The area of the smaller rectangle is:
\[
\text{Area}_{\text{small}} = \text{Length}_{\text{small}} \times \text{Width}_{\text{small}} = 6 \times 4 = 24 \, \text{square units}
\]
4. **Area of the shaded region (the frame)**:
The area of the shaded region is the area of the larger rectangle minus the area of the smaller rectangle:
\[
\text{Area}_{\text{shaded}} = \text{Area}_{\text{large}} - \text{Area}_{\text{small}} = 120 - 24 = 96 \, \text{square units}
\]
Therefore, the area of the shaded region is \(\boxed{96}\) square units. If there are any specific constraints or misunderstandings in the problem that make this answer seem incorrect, please provide more details.
1. **Area of the larger rectangle**:
The larger rectangle has dimensions 12 units by 10 units, so its area is:
\[
\text{Area}_{\text{large}} = \text{length} \times \text{width} = 12 \times 10 = 120 \, \text{square units}
\]
2. **Dimensions of the smaller rectangle**:
Since the frame is 3 units wide, we need to subtract 3 units from each side of the larger rectangle to find the dimensions of the smaller rectangle.
- The width of the smaller rectangle is:
\[
\text{Width}_{\text{small}} = \text{Width}_{\text{large}} - 2 \times \text{frame width} = 10 - 2 \times 3 = 10 - 6 = 4 \, \text{units}
\]
- The length of the smaller rectangle is:
\[
\text{Length}_{\text{small}} = \text{Length}_{\text{large}} - 2 \times \text{frame width} = 12 - 2 \times 3 = 12 - 6 = 6 \, \text{units}
\]
3. **Area of the smaller rectangle**:
The area of the smaller rectangle is:
\[
\text{Area}_{\text{small}} = \text{Length}_{\text{small}} \times \text{Width}_{\text{small}} = 6 \times 4 = 24 \, \text{square units}
\]
4. **Area of the shaded region (the frame)**:
The area of the shaded region is the area of the larger rectangle minus the area of the smaller rectangle:
\[
\text{Area}_{\text{shaded}} = \text{Area}_{\text{large}} - \text{Area}_{\text{small}} = 120 - 24 = 96 \, \text{square units}
\]
Therefore, the area of the shaded region is \(\boxed{96}\) square units. If there are any specific constraints or misunderstandings in the problem that make this answer seem incorrect, please provide more details.