Question

select the correct answer.

a company makes the forks for forklift trucks. each fork has the dimensions shown. the top of each fork is painted. to the nearest square millimeter, how much area is painted on the top of each fork?

\\(83,425\text{ mm}^2\\)
\\(81,000\text{ mm}^2\\)
\\(2,425\text{ mm}^2\\)
\\(84,118\text{ mm}^2\\)

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Area of Polygons"
],
"new_concepts": [
"Area of Composite Figures",
"Trigonometric Area Formula"
],
"current_concepts": [
"Area of Polygons",
"Area of Composite Figures",
"Trigonometric Area Formula"
]
}
</pre_analysis>

<reasoning>

Partition the shape into a rectangle and a trapezoid

\[

$$\begin{aligned} &\text{Total Area} = A_{\text{rectangle}} + A_{\text{trapezoid}}\\ &A_{\text{rectangle}} = 900 \text{ mm} \times 90 \text{ mm} = 81,000 \text{ mm}^2 \end{aligned}$$

\]

Calculate the area of the trapezoidal tip

\[

$$\begin{aligned} &h_{\text{trapezoid}} = 40 \cdot \sin(60^\circ) = 40 \cdot \frac{\sqrt{3}}{2} = 20\sqrt{3} \approx 34.641 \text{ mm}\\ &a = 90 \text{ mm},\quad b = 50 \text{ mm}\\ &A_{\text{trapezoid}} = \frac{a + b}{2} \cdot h_{\text{trapezoid}} = \frac{90 + 50}{2} \cdot 20\sqrt{3} = 1400\sqrt{3} \approx 2424.87 \text{ mm}^2 \end{aligned}$$

\]

Sum the areas to find the total painted area

\[

$$\begin{aligned} &\text{Total Area} = 81,000 + 1400\sqrt{3} \approx 81,000 + 2424.87 = 83,424.87 \text{ mm}^2\\ &\text{Total Area} \approx 83,425 \text{ mm}^2 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) \(83,425\text{ mm}^2\)</mcq-correct>
<mcq-option>(B) \(81,000\text{ mm}^2\)</mcq-option>
<mcq-option>(C) \(2,425\text{ mm}^2\)</mcq-option>
<mcq-option>(D) \(84,118\text{ mm}^2\)</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Area of Composite Figures"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Area of Polygons"
],
"new_concepts": [
"Area of Composite Figures",
"Trigonometric Area Formula"
],
"current_concepts": [
"Area of Polygons",
"Area of Composite Figures",
"Trigonometric Area Formula"
]
}
</pre_analysis>

<reasoning>

Partition the shape into a rectangle and a trapezoid

\[

$$\begin{aligned} &\text{Total Area} = A_{\text{rectangle}} + A_{\text{trapezoid}}\\ &A_{\text{rectangle}} = 900 \text{ mm} \times 90 \text{ mm} = 81,000 \text{ mm}^2 \end{aligned}$$

\]

Calculate the area of the trapezoidal tip

\[

$$\begin{aligned} &h_{\text{trapezoid}} = 40 \cdot \sin(60^\circ) = 40 \cdot \frac{\sqrt{3}}{2} = 20\sqrt{3} \approx 34.641 \text{ mm}\\ &a = 90 \text{ mm},\quad b = 50 \text{ mm}\\ &A_{\text{trapezoid}} = \frac{a + b}{2} \cdot h_{\text{trapezoid}} = \frac{90 + 50}{2} \cdot 20\sqrt{3} = 1400\sqrt{3} \approx 2424.87 \text{ mm}^2 \end{aligned}$$

\]

Sum the areas to find the total painted area

\[

$$\begin{aligned} &\text{Total Area} = 81,000 + 1400\sqrt{3} \approx 81,000 + 2424.87 = 83,424.87 \text{ mm}^2\\ &\text{Total Area} \approx 83,425 \text{ mm}^2 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) \(83,425\text{ mm}^2\)</mcq-correct>
<mcq-option>(B) \(81,000\text{ mm}^2\)</mcq-option>
<mcq-option>(C) \(2,425\text{ mm}^2\)</mcq-option>
<mcq-option>(D) \(84,118\text{ mm}^2\)</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Area of Composite Figures"
]
}
</post_analysis>