Question

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the surface area of a cone is given by the formula \\(s = \pi r l + \pi r^2\\). solve the formula for \\(l\\).

\\(l = s - r^2\\)
\\(l = s + r^2\\)
\\(l = \frac{s}{\pi r} - r\\)
\\(l = \frac{s}{\pi r} + r\\)

Explanation:

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<reasoning>

State the given equation

We begin with the formula for the surface area of a cone:
\[S = \pi r l + \pi r^2\]

Isolate the term containing l

Subtract \(\pi r^2\) from both sides of the equation:
\[S - \pi r^2 = \pi r l\]

Solve for l

Divide both sides by \(\pi r\) to isolate \(l\):
\[l = \frac{S - \pi r^2}{\pi r}\]

Simplify the expression

Divide each term in the numerator by the denominator:
\[l = \frac{S}{\pi r} - \frac{\pi r^2}{\pi r}\]
\[l = \frac{S}{\pi r} - r\]
</reasoning>

<answer>
<mcq-option>(A) \(l = S - r^2\)</mcq-option>
<mcq-option>(B) \(l = S + r^2\)</mcq-option>
<mcq-correct>(C) \(l = \frac{S}{\pi r} - r\)</mcq-correct>
<mcq-option>(D) \(l = \frac{S}{\pi r} + r\)</mcq-option>
</answer>

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Answer:

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<reasoning>

State the given equation

We begin with the formula for the surface area of a cone:
\[S = \pi r l + \pi r^2\]

Isolate the term containing l

Subtract \(\pi r^2\) from both sides of the equation:
\[S - \pi r^2 = \pi r l\]

Solve for l

Divide both sides by \(\pi r\) to isolate \(l\):
\[l = \frac{S - \pi r^2}{\pi r}\]

Simplify the expression

Divide each term in the numerator by the denominator:
\[l = \frac{S}{\pi r} - \frac{\pi r^2}{\pi r}\]
\[l = \frac{S}{\pi r} - r\]
</reasoning>

<answer>
<mcq-option>(A) \(l = S - r^2\)</mcq-option>
<mcq-option>(B) \(l = S + r^2\)</mcq-option>
<mcq-correct>(C) \(l = \frac{S}{\pi r} - r\)</mcq-correct>
<mcq-option>(D) \(l = \frac{S}{\pi r} + r\)</mcq-option>
</answer>

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