Question

1. 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

1. a square has side length 5 cm and is enlarged by a scale factor of 3. 4 marks

Explanation:

<pre_analysis>
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"skills_matched": [
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"explain_then_answer",
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],
"needs_drawing": false,
"concepts_used": [
"Significant Figures",
"Surface Area and Volume"
],
"new_concepts": [
"Scale Factor"
],
"current_concepts": [
"Significant Figures",
"Surface Area and Volume",
"Scale Factor"
]
}
</pre_analysis>

<reasoning>

计算圆柱体的体积

Using the Surface Area and Volume knowledge point
\[

$$\begin{aligned} r &= 6\text{ cm},\quad h = 14\text{ cm}\\ V &= \pi r^2 h = \pi \times 6^2 \times 14 = 504\pi\text{ cm}^3\\ V &\approx 1583.36\text{ cm}^3 \end{aligned}$$

\]

计算圆柱体的总表面积

Using the Surface Area and Volume knowledge point
\[

$$\begin{aligned} A &= 2\pi r^2 + 2\pi r h = 2\pi \times 6^2 + 2\pi \times 6 \times 14\\ A &= 72\pi + 168\pi = 240\pi\text{ cm}^2\\ A &\approx 753.98\text{ cm}^2 \end{aligned}$$

\]

保留三位有效数字

Using the Significant Figures knowledge point
\[

$$\begin{aligned} V &\approx 1580\text{ cm}^3 \quad (3\text{ s.f.})\\ A &\approx 754\text{ cm}^2 \quad (3\text{ s.f.}) \end{aligned}$$

\]

分析正方形的放大

正方形的边长为 \(5\text{ cm}\)，按比例因子 \(3\) 进行放大。
放大后的新边长为：
\[
5\text{ cm} \times 3 = 15\text{ cm}
\]
</reasoning>

<answer>

Question 15

圆柱体的体积和总表面积（保留至 3 位有效数字）分别为：

• 体积：<blank>\(1580\text{ cm}^3\)</blank>
• 总表面积：<blank>\(754\text{ cm}^2\)</blank>

Question 17

正方形放大后的新边长为：

• 新边长：<blank>\(15\text{ cm}\)</blank>

</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Geometry",
"Surface Area and Volume"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 2,
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"step_cot",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Significant Figures",
"Surface Area and Volume"
],
"new_concepts": [
"Scale Factor"
],
"current_concepts": [
"Significant Figures",
"Surface Area and Volume",
"Scale Factor"
]
}
</pre_analysis>

<reasoning>

计算圆柱体的体积

Using the Surface Area and Volume knowledge point
\[

$$\begin{aligned} r &= 6\text{ cm},\quad h = 14\text{ cm}\\ V &= \pi r^2 h = \pi \times 6^2 \times 14 = 504\pi\text{ cm}^3\\ V &\approx 1583.36\text{ cm}^3 \end{aligned}$$

\]

计算圆柱体的总表面积

Using the Surface Area and Volume knowledge point
\[

$$\begin{aligned} A &= 2\pi r^2 + 2\pi r h = 2\pi \times 6^2 + 2\pi \times 6 \times 14\\ A &= 72\pi + 168\pi = 240\pi\text{ cm}^2\\ A &\approx 753.98\text{ cm}^2 \end{aligned}$$

\]

保留三位有效数字

Using the Significant Figures knowledge point
\[

$$\begin{aligned} V &\approx 1580\text{ cm}^3 \quad (3\text{ s.f.})\\ A &\approx 754\text{ cm}^2 \quad (3\text{ s.f.}) \end{aligned}$$

\]

分析正方形的放大

正方形的边长为 \(5\text{ cm}\)，按比例因子 \(3\) 进行放大。
放大后的新边长为：
\[
5\text{ cm} \times 3 = 15\text{ cm}
\]
</reasoning>

<answer>

Question 15

圆柱体的体积和总表面积（保留至 3 位有效数字）分别为：

• 体积：<blank>\(1580\text{ cm}^3\)</blank>
• 总表面积：<blank>\(754\text{ cm}^2\)</blank>

Question 17

正方形放大后的新边长为：

• 新边长：<blank>\(15\text{ cm}\)</blank>

</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Geometry",
"Surface Area and Volume"
]
}
</post_analysis>