QUESTION IMAGE
Question
- find the area of a circle with radius 7 cm. round your answer to 3 s.f. 2 marks
- a semicircle has diameter 12 m. calculate its area to 1 d.p. 2 marks
- the shape is a 10 cm \\(\times\\) 8 cm rectangle with a quarter circle (r = 4 cm) removed from one corner. find the shaded area to 3 s.f. 3 marks
- a running track has two straight sections each 80 m long and two semicircular ends each with diameter 50 m. find the total enclosed area to 3 s.f. draw it out 3 marks
section d - surface area & volume 12 marks
- a cuboid has length 9 cm, width 5 cm, and height 3 cm. 4 marks
(a) calculate the volume.
(b) calculate the total surface area.
- a cylinder has radius 6 cm and height 14 cm. calculate its volume and total surface area to 3 s.f. 4 marks
section e - scale factor 4 marks
- a square has side length 5 cm and is enlarged by a scale factor of 3. 4 marks
(a) what is the side length of the enlarged square?
(b) find the area of both the original and the enlarged square.
<pre_analysis>
{
"quality": "clear",
"question_count": 9,
"skills_matched": [
"step_cot",
"explain_then_answer",
"tutor"
],
"needs_drawing": false,
"concepts_used": [
"Significant Figures"
],
"new_concepts": [
"Area of Circles",
"Composite Shapes",
"Surface Area and Volume"
],
"current_concepts": [
"Significant Figures",
"Area of Circles",
"Composite Shapes",
"Surface Area and Volume",
"Scale Factor"
]
}
</pre_analysis>
<reasoning>
计算圆的面积
Using the Significant Figures knowledge point
已知圆的半径 \(r = 7\text{ cm}\)。
圆的面积公式为：
\[A = \pi r^2 = \pi \times 7^2 = 49\pi \approx 153.938\text{ cm}^2\]
保留 3 位有效数字（Significant Figures）为 \(154\text{ cm}^2\)。
计算半圆的面积
已知半圆直径 \(d = 12\text{ m}\)，则半径 \(r = 6\text{ m}\)。
半圆面积公式为：
\[A = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi \times 6^2 = 18\pi \approx 56.548\text{ m}^2\]
保留 1 位小数为 \(56.5\text{ m}^2\)。
计算组合图形的阴影面积
矩形面积为：
\[A_{\text{rect}} = 10 \times 8 = 80\text{ cm}^2\]
空白四分之一圆的半径 \(r = 4\text{ cm}\)，其面积为：
\[A_{\text{quarter}} = \frac{1}{4} \pi r^2 = \frac{1}{4} \pi \times 4^2 = 4\pi \approx 12.566\text{ cm}^2\]
阴影部分面积为：
\[A_{\text{shaded}} = 80 - 4\pi \approx 80 - 12.566 = 67.434\text{ cm}^2\]
保留 3 位有效数字（Significant Figures）为 \(67.4\text{ cm}^2\)。
计算跑道总面积
跑道由一个矩形和两个半圆（即一个整圆）组成。
矩形长 \(l = 80\text{ m}\)，宽等于半圆直径 \(d = 50\text{ m}\)。
矩形面积为：
\[A_{\text{rect}} = 80 \times 50 = 4000\text{ m}^2\]
两端半圆拼成一个直径为 \(50\text{ m}\)（半径 \(r = 25\text{ m}\)）的整圆，其面积为：
\[A_{\text{circle}} = \pi r^2 = \pi \times 25^2 = 625\pi \approx 1963.495\text{ m}^2\]
总面积为：
\[A_{\text{total}} = 4000 + 1963.495 = 5963.495\text{ m}^2\]
保留 3 位有效数字（Significant Figures）为 \(5960\text{ m}^2\)。
计算长方体的体积与表面积
已知长方体长 \(l = 9\text{ cm}\)，宽 \(w = 5\text{ cm}\)，高 \(h = 3\text{ cm}\)。
(a) 体积公式为：
\[V = l \times w \times h = 9 \times 5 \times 3 = 135\text{ cm}^3\]
(b) 表面积公式为：
\[A = 2(lw + lh + wh) = 2(9\times5 + 9\times3 + 5\times3) = 2(45 + 27 + 15) = 2 \times 87 = 174\text{ cm}^2\]
计算圆柱的体积与表面积
Using the Significant Figures knowledge point
已知圆柱底面半径 \(r = 6\text{ cm}\)，高 \(h = 14\text{ cm}\)。
体积公式为：
\[V = \pi r^2 h = \pi \times 6^2 \times 14 = 504\pi \approx 1583.36\text{ cm}^3\]
保留 3 位有效数字（Significant Figures）为 \(1580\text{ cm}^3\)。
表面积公式为：
\[A = 2\pi r^2 + 2\pi r h = 2\pi \times 6^2 + 2\pi \times 6 \times 14 = 72\pi + 168\pi = 240\pi \approx 753.98\text{ cm}^2\]
保留 3 位有效数字（Significant Figures）为 \(754\text{ cm}^2\)。
计算放大正方形的边长与面积
已知原正方形边长为 \(5\text{ cm}\)，放大比例尺（比例因子）为 \(3\)。
(a) 放大后的正方形边长为：
\[a_{\text{new}} = 5 \times 3 = 15\text{ cm}\]
(b) 原正方形面积为：
\[A_{\text{orig}} = 5^2 = 25\text{ cm}^2\]
放大后的正方形面积为：
\[A_{\text{new}} = 15^2 = 225\text{ cm}^2\]
</reasoning>
<answer>
Question 10
\(154\text{ cm}^2\)
Question 11
\(56.5\text{ m}^2\)
Question 12
\(67.4\text{ cm}^2\)
Question 13
\(5960\text{ m}^2\)
Question 14
(a) \(135\text{ cm}^3\)
(b) \(174\text{ cm}^2\)
Question 15
体积：\(1580\text{ cm}^3\)
总表面积：\(754\text{ cm}^2\)
Question 17
(a) \(15\text{ cm}\)
(b) 原正方形面积：\(25\text{ cm}^2\)，放大后正方形面积：\(225\text{ cm}^2\)
</answer>
<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Geometry",
"Area and Volume"
]
}
</post_analysis>
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<pre_analysis>
{
"quality": "clear",
"question_count": 9,
"skills_matched": [
"step_cot",
"explain_then_answer",
"tutor"
],
"needs_drawing": false,
"concepts_used": [
"Significant Figures"
],
"new_concepts": [
"Area of Circles",
"Composite Shapes",
"Surface Area and Volume"
],
"current_concepts": [
"Significant Figures",
"Area of Circles",
"Composite Shapes",
"Surface Area and Volume",
"Scale Factor"
]
}
</pre_analysis>
<reasoning>
计算圆的面积
Using the Significant Figures knowledge point
已知圆的半径 \(r = 7\text{ cm}\)。
圆的面积公式为：
\[A = \pi r^2 = \pi \times 7^2 = 49\pi \approx 153.938\text{ cm}^2\]
保留 3 位有效数字（Significant Figures）为 \(154\text{ cm}^2\)。
计算半圆的面积
已知半圆直径 \(d = 12\text{ m}\)，则半径 \(r = 6\text{ m}\)。
半圆面积公式为：
\[A = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi \times 6^2 = 18\pi \approx 56.548\text{ m}^2\]
保留 1 位小数为 \(56.5\text{ m}^2\)。
计算组合图形的阴影面积
矩形面积为：
\[A_{\text{rect}} = 10 \times 8 = 80\text{ cm}^2\]
空白四分之一圆的半径 \(r = 4\text{ cm}\)，其面积为：
\[A_{\text{quarter}} = \frac{1}{4} \pi r^2 = \frac{1}{4} \pi \times 4^2 = 4\pi \approx 12.566\text{ cm}^2\]
阴影部分面积为：
\[A_{\text{shaded}} = 80 - 4\pi \approx 80 - 12.566 = 67.434\text{ cm}^2\]
保留 3 位有效数字（Significant Figures）为 \(67.4\text{ cm}^2\)。
计算跑道总面积
跑道由一个矩形和两个半圆（即一个整圆）组成。
矩形长 \(l = 80\text{ m}\)，宽等于半圆直径 \(d = 50\text{ m}\)。
矩形面积为：
\[A_{\text{rect}} = 80 \times 50 = 4000\text{ m}^2\]
两端半圆拼成一个直径为 \(50\text{ m}\)（半径 \(r = 25\text{ m}\)）的整圆，其面积为：
\[A_{\text{circle}} = \pi r^2 = \pi \times 25^2 = 625\pi \approx 1963.495\text{ m}^2\]
总面积为：
\[A_{\text{total}} = 4000 + 1963.495 = 5963.495\text{ m}^2\]
保留 3 位有效数字（Significant Figures）为 \(5960\text{ m}^2\)。
计算长方体的体积与表面积
已知长方体长 \(l = 9\text{ cm}\)，宽 \(w = 5\text{ cm}\)，高 \(h = 3\text{ cm}\)。
(a) 体积公式为：
\[V = l \times w \times h = 9 \times 5 \times 3 = 135\text{ cm}^3\]
(b) 表面积公式为：
\[A = 2(lw + lh + wh) = 2(9\times5 + 9\times3 + 5\times3) = 2(45 + 27 + 15) = 2 \times 87 = 174\text{ cm}^2\]
计算圆柱的体积与表面积
Using the Significant Figures knowledge point
已知圆柱底面半径 \(r = 6\text{ cm}\)，高 \(h = 14\text{ cm}\)。
体积公式为：
\[V = \pi r^2 h = \pi \times 6^2 \times 14 = 504\pi \approx 1583.36\text{ cm}^3\]
保留 3 位有效数字（Significant Figures）为 \(1580\text{ cm}^3\)。
表面积公式为：
\[A = 2\pi r^2 + 2\pi r h = 2\pi \times 6^2 + 2\pi \times 6 \times 14 = 72\pi + 168\pi = 240\pi \approx 753.98\text{ cm}^2\]
保留 3 位有效数字（Significant Figures）为 \(754\text{ cm}^2\)。
计算放大正方形的边长与面积
已知原正方形边长为 \(5\text{ cm}\)，放大比例尺（比例因子）为 \(3\)。
(a) 放大后的正方形边长为：
\[a_{\text{new}} = 5 \times 3 = 15\text{ cm}\]
(b) 原正方形面积为：
\[A_{\text{orig}} = 5^2 = 25\text{ cm}^2\]
放大后的正方形面积为：
\[A_{\text{new}} = 15^2 = 225\text{ cm}^2\]
</reasoning>
<answer>
Question 10
\(154\text{ cm}^2\)
Question 11
\(56.5\text{ m}^2\)
Question 12
\(67.4\text{ cm}^2\)
Question 13
\(5960\text{ m}^2\)
Question 14
(a) \(135\text{ cm}^3\)
(b) \(174\text{ cm}^2\)
Question 15
体积：\(1580\text{ cm}^3\)
总表面积：\(754\text{ cm}^2\)
Question 17
(a) \(15\text{ cm}\)
(b) 原正方形面积：\(25\text{ cm}^2\)，放大后正方形面积：\(225\text{ cm}^2\)
</answer>
<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Geometry",
"Area and Volume"
]
}
</post_analysis>