Question

a circular disc has an area that measures between 1,650 and 1,700 square inches.

what could be the length of its radius? use 3.14 for \\(\pi\\).

23 in.
24 in.
25 in.
26 in.

Explanation:

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"concepts_used": [
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],
"new_concepts": [
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"current_concepts": [
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<reasoning>

Set up the inequality for the area of the circle

\[
1650 < \pi r^2 < 1700
\]
\[
1650 < 3.14 \cdot r^2 < 1700
\]

Solve for the range of the radius

\[
\frac{1650}{3.14} < r^2 < \frac{1700}{3.14}
\]
\[
525.48 < r^2 < 541.40
\]
\[
\sqrt{525.48} < r < \sqrt{541.40}
\]
\[
22.92 < r < 23.27
\]

Evaluate the given options

\[

$$\begin{aligned} &\text{For } r = 23: \quad 22.92 < 23 < 23.27 \quad (\text{True}) \\ &\text{For } r = 24: \quad 24 > 23.27 \quad (\text{False}) \\ &\text{For } r = 25: \quad 25 > 23.27 \quad (\text{False}) \\ &\text{For } r = 26: \quad 26 > 23.27 \quad (\text{False}) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) 23 in.</mcq-correct>
<mcq-option>(B) 24 in.</mcq-option>
<mcq-option>(C) 25 in.</mcq-option>
<mcq-option>(D) 26 in.</mcq-option>
</answer>

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]
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Answer:

<pre_analysis>
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"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
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],
"new_concepts": [
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]
}
</pre_analysis>

<reasoning>

Set up the inequality for the area of the circle

\[
1650 < \pi r^2 < 1700
\]
\[
1650 < 3.14 \cdot r^2 < 1700
\]

Solve for the range of the radius

\[
\frac{1650}{3.14} < r^2 < \frac{1700}{3.14}
\]
\[
525.48 < r^2 < 541.40
\]
\[
\sqrt{525.48} < r < \sqrt{541.40}
\]
\[
22.92 < r < 23.27
\]

Evaluate the given options

\[

$$\begin{aligned} &\text{For } r = 23: \quad 22.92 < 23 < 23.27 \quad (\text{True}) \\ &\text{For } r = 24: \quad 24 > 23.27 \quad (\text{False}) \\ &\text{For } r = 25: \quad 25 > 23.27 \quad (\text{False}) \\ &\text{For } r = 26: \quad 26 > 23.27 \quad (\text{False}) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) 23 in.</mcq-correct>
<mcq-option>(B) 24 in.</mcq-option>
<mcq-option>(C) 25 in.</mcq-option>
<mcq-option>(D) 26 in.</mcq-option>
</answer>

<post_analysis>
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"question_type": "Multiple Choice",
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"Mathematics",
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