Question

select the correct answer.

an engineering firm designs a custom hexagonal screw for a computer board. a sketch of the top of the screw is shown. what is the area of the screw head?

(a) 62 \text{ mm}^2
(b) 187 \text{ mm}^2
(c) 36 \text{ mm}^2
(d) 108 \text{ mm}^2

Explanation:

<pre_analysis>
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<reasoning>

Divide the hexagon into simpler geometric shapes

\[

$$\begin{aligned} &\text{Vertices of the hexagon: } (0, 6), (6, 3), (6, -3), (0, -6), (-6, -3), (-6, 3)\\ &\text{Divide the hexagon vertically along the } y\text{-axis into two symmetric pentagons, or horizontally into:}\\ &\text{1. A central rectangle with vertices } (-6, 3), (6, 3), (6, -3), (-6, -3)\\ &\text{2. A top triangle with vertices } (-6, 3), (6, 3), (0, 6)\\ &\text{3. A bottom triangle with vertices } (-6, -3), (6, -3), (0, -6) \end{aligned}$$

\]

Calculate the area of each component

\[

$$\begin{aligned} &\text{Area of central rectangle: } A_{\text{rect}} = \text{width} \times \text{height} = (6 - (-6)) \times (3 - (-3)) = 12 \times 6 = 72\text{ mm}^2\\ &\text{Area of top triangle: } A_{\text{top}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 12 \times (6 - 3) = \frac{1}{2} \times 12 \times 3 = 18\text{ mm}^2\\ &\text{Area of bottom triangle: } A_{\text{bottom}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 12 \times (-3 - (-6)) = \frac{1}{2} \times 12 \times 3 = 18\text{ mm}^2 \end{aligned}$$

\]

Sum the areas to find the total area

\[

$$\begin{aligned} &A_{\text{total}} = A_{\text{rect}} + A_{\text{top}} + A_{\text{bottom}} = 72 + 18 + 18 = 108\text{ mm}^2 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>\(62\text{ mm}^2\)</mcq-option>
<mcq-option>\(187\text{ mm}^2\)</mcq-option>
<mcq-option>\(36\text{ mm}^2\)</mcq-option>
<mcq-correct>\(108\text{ mm}^2\)</mcq-correct>
</answer>

<post_analysis>
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"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Area of Polygons"
]
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</post_analysis>

Answer:

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</pre_analysis>

<reasoning>

Divide the hexagon into simpler geometric shapes

\[

$$\begin{aligned} &\text{Vertices of the hexagon: } (0, 6), (6, 3), (6, -3), (0, -6), (-6, -3), (-6, 3)\\ &\text{Divide the hexagon vertically along the } y\text{-axis into two symmetric pentagons, or horizontally into:}\\ &\text{1. A central rectangle with vertices } (-6, 3), (6, 3), (6, -3), (-6, -3)\\ &\text{2. A top triangle with vertices } (-6, 3), (6, 3), (0, 6)\\ &\text{3. A bottom triangle with vertices } (-6, -3), (6, -3), (0, -6) \end{aligned}$$

\]

Calculate the area of each component

\[

$$\begin{aligned} &\text{Area of central rectangle: } A_{\text{rect}} = \text{width} \times \text{height} = (6 - (-6)) \times (3 - (-3)) = 12 \times 6 = 72\text{ mm}^2\\ &\text{Area of top triangle: } A_{\text{top}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 12 \times (6 - 3) = \frac{1}{2} \times 12 \times 3 = 18\text{ mm}^2\\ &\text{Area of bottom triangle: } A_{\text{bottom}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 12 \times (-3 - (-6)) = \frac{1}{2} \times 12 \times 3 = 18\text{ mm}^2 \end{aligned}$$

\]

Sum the areas to find the total area

\[

$$\begin{aligned} &A_{\text{total}} = A_{\text{rect}} + A_{\text{top}} + A_{\text{bottom}} = 72 + 18 + 18 = 108\text{ mm}^2 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>\(62\text{ mm}^2\)</mcq-option>
<mcq-option>\(187\text{ mm}^2\)</mcq-option>
<mcq-option>\(36\text{ mm}^2\)</mcq-option>
<mcq-correct>\(108\text{ mm}^2\)</mcq-correct>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Area of Polygons"
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