Question

exit ticket ma.gr.1.1

area of composite shapes

find the area

1)
2)
3)
4)
5)
6)

Explanation:

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<reasoning>

Decompose shape 1 into rectangles

The U-shape has a base width of \(16\text{ in}\) and total height of \(16\text{ in}\).
The inner cutout has a width of \(8\text{ in}\) and depth of \(10\text{ in}\).
We can split this into a large outer rectangle minus the inner cutout rectangle:
\[
\text{Area}_1 = (16 \times 16) - (8 \times 10) = 256 - 80 = 176\text{ in}^2
\]

Decompose shape 2 into rectangles

The shape is an L-like polygon.
The top horizontal side is \(10\text{ m}\) and the right vertical side is \(10\text{ m}\).
The inner cutout has horizontal step \(4\text{ m}\) and vertical step \(4\text{ m}\).
This means the bottom width is \(10 - 4 = 6\text{ m}\) and the left height is \(10 - 4 = 6\text{ m}\).
We can split this into a main \(10 \times 6\) rectangle and a bottom-right \(6 \times 4\) rectangle:
\[
\text{Area}_2 = (10 \times 6) + (6 \times 4) = 60 + 24 = 84\text{ m}^2
\]

Decompose shape 3 into rectangles

The shape consists of a top square and a bottom rectangle.
The top square has dimensions \(7\text{ m} \times 7\text{ m}\).
The bottom rectangle has dimensions \(3\text{ m} \times 5\text{ m}\).
We sum their areas:
\[
\text{Area}_3 = (7 \times 7) + (3 \times 5) = 49 + 15 = 64\text{ m}^2
\]

Decompose shape 4 into rectangles

The shape is an L-shape.
The top horizontal side is \(11\text{ ft}\) and the right vertical side is \(13\text{ ft}\).
The inner vertical step is \(8\text{ ft}\) and the inner horizontal step is \(6\text{ ft}\).
We can split this vertically into a left rectangle of \(5\text{ ft} \times 5\text{ ft}\) (since \(11 - 6 = 5\) and \(13 - 8 = 5\)) and a right rectangle of \(6\text{ ft} \times 13\text{ ft}\):
\[
\text{Area}_4 = (5 \times 5) + (6 \times 13) = 25 + 78 = 103\text{ ft}^2
\]

Decompose shape 5 into rectangles

The shape consists of a top rectangle and a bottom rectangle.
The top rectangle has dimensions \(15\text{ cm} \times 12\text{ cm}\).
The bottom rectangle has dimensions \(6\text{ cm} \times 8\text{ cm}\).
We sum their areas:
\[
\text{Area}_5 = (15 \times 12) + (6 \times 8) = 180 + 48 = 228\text{ cm}^2
\]

Decompose shape 6 into rectangles

The U-shape has a base width of \(14\text{ ft}\) and total height of \(11\text{ ft}\).
The inner cutout has a width of \(9\text{ ft}\) and depth of \(7\text{ ft}\).
We subtract the cutout area from the outer rectangle area:
\[
\text{Area}_6 = (14 \times 11) - (9 \times 7) = 154 - 63 = 91\text{ ft}^2
\]
</reasoning>

<answer>

No.	Answer
2	\(84\text{ m}^2\)
3	\(64\text{ m}^2\)
4	\(103\text{ ft}^2\)
5	\(228\text{ cm}^2\)
6	\(91\text{ ft}^2\)

</answer>

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"question_type": "Free Response",
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"Area of Composite Shapes"
]
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</post_analysis>

Answer:

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</pre_analysis>

<reasoning>

Decompose shape 1 into rectangles

The U-shape has a base width of \(16\text{ in}\) and total height of \(16\text{ in}\).
The inner cutout has a width of \(8\text{ in}\) and depth of \(10\text{ in}\).
We can split this into a large outer rectangle minus the inner cutout rectangle:
\[
\text{Area}_1 = (16 \times 16) - (8 \times 10) = 256 - 80 = 176\text{ in}^2
\]

Decompose shape 2 into rectangles

The shape is an L-like polygon.
The top horizontal side is \(10\text{ m}\) and the right vertical side is \(10\text{ m}\).
The inner cutout has horizontal step \(4\text{ m}\) and vertical step \(4\text{ m}\).
This means the bottom width is \(10 - 4 = 6\text{ m}\) and the left height is \(10 - 4 = 6\text{ m}\).
We can split this into a main \(10 \times 6\) rectangle and a bottom-right \(6 \times 4\) rectangle:
\[
\text{Area}_2 = (10 \times 6) + (6 \times 4) = 60 + 24 = 84\text{ m}^2
\]

Decompose shape 3 into rectangles

The shape consists of a top square and a bottom rectangle.
The top square has dimensions \(7\text{ m} \times 7\text{ m}\).
The bottom rectangle has dimensions \(3\text{ m} \times 5\text{ m}\).
We sum their areas:
\[
\text{Area}_3 = (7 \times 7) + (3 \times 5) = 49 + 15 = 64\text{ m}^2
\]

Decompose shape 4 into rectangles

The shape is an L-shape.
The top horizontal side is \(11\text{ ft}\) and the right vertical side is \(13\text{ ft}\).
The inner vertical step is \(8\text{ ft}\) and the inner horizontal step is \(6\text{ ft}\).
We can split this vertically into a left rectangle of \(5\text{ ft} \times 5\text{ ft}\) (since \(11 - 6 = 5\) and \(13 - 8 = 5\)) and a right rectangle of \(6\text{ ft} \times 13\text{ ft}\):
\[
\text{Area}_4 = (5 \times 5) + (6 \times 13) = 25 + 78 = 103\text{ ft}^2
\]

Decompose shape 5 into rectangles

The shape consists of a top rectangle and a bottom rectangle.
The top rectangle has dimensions \(15\text{ cm} \times 12\text{ cm}\).
The bottom rectangle has dimensions \(6\text{ cm} \times 8\text{ cm}\).
We sum their areas:
\[
\text{Area}_5 = (15 \times 12) + (6 \times 8) = 180 + 48 = 228\text{ cm}^2
\]

Decompose shape 6 into rectangles

The U-shape has a base width of \(14\text{ ft}\) and total height of \(11\text{ ft}\).
The inner cutout has a width of \(9\text{ ft}\) and depth of \(7\text{ ft}\).
We subtract the cutout area from the outer rectangle area:
\[
\text{Area}_6 = (14 \times 11) - (9 \times 7) = 154 - 63 = 91\text{ ft}^2
\]
</reasoning>

<answer>

No.	Answer
2	\(84\text{ m}^2\)
3	\(64\text{ m}^2\)
4	\(103\text{ ft}^2\)
5	\(228\text{ cm}^2\)
6	\(91\text{ ft}^2\)

</answer>

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"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Geometry",
"Area of Composite Shapes"
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