Question

using a rewritten formula

the formula for the slant height of a cone is \\(l = \frac{s - \pi r^2}{\pi r}\\), where \\(s\\) is surface area of the cone. use the formula to find the slant height, \\(l\\), of a cone with a surface area of \\(1500\pi\text{ ft}^2\\) and a radius of \\(15\text{ ft}\\).
\\(l =\\) \\(\text{ft}\\)

Explanation:

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<reasoning>

Identify given values and formula

\[
l = \frac{S - \pi r^2}{\pi r},\quad S = 1500\pi\text{ ft}^2,\quad r = 15\text{ ft}
\]

Substitute values into the formula

\[
l = \frac{1500\pi - \pi (15)^2}{\pi (15)}
\]

Simplify the expression

\[
l = \frac{1500\pi - 225\pi}{15\pi} = \frac{1275\pi}{15\pi} = \frac{1275}{15} = 85
\]
</reasoning>

<answer>
The formula for the slant height of a cone is \(l = \frac{S - \pi r^2}{\pi r}\), where \(S\) is surface area of the cone. Use the formula to find the slant height, \(l\), of a cone with a surface area of \(1500\pi\text{ ft}^2\) and a radius of \(15\text{ ft}\).
\(l =\) <blank>85</blank> \(\text{ft}\)
</answer>

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"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
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"Literal Equations"
]
}
</post_analysis>

Answer:

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"new_concepts": [
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<reasoning>

Identify given values and formula

\[
l = \frac{S - \pi r^2}{\pi r},\quad S = 1500\pi\text{ ft}^2,\quad r = 15\text{ ft}
\]

Substitute values into the formula

\[
l = \frac{1500\pi - \pi (15)^2}{\pi (15)}
\]

Simplify the expression

\[
l = \frac{1500\pi - 225\pi}{15\pi} = \frac{1275\pi}{15\pi} = \frac{1275}{15} = 85
\]
</reasoning>

<answer>
The formula for the slant height of a cone is \(l = \frac{S - \pi r^2}{\pi r}\), where \(S\) is surface area of the cone. Use the formula to find the slant height, \(l\), of a cone with a surface area of \(1500\pi\text{ ft}^2\) and a radius of \(15\text{ ft}\).
\(l =\) <blank>85</blank> \(\text{ft}\)
</answer>

<post_analysis>
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"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Literal Equations"
]
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