Question

the radius of the cone is 7 in and y = 25 in. what is the volume of the cone in terms of \\(\pi\\)? (1 point)

\\(392\pi\text{ in}^3\\)
\\(408\pi\text{ in}^3\\)
\\(112\pi\text{ in}^3\\)
\\(117\pi\text{ in}^3\\)

Explanation:

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<reasoning>

Identify the given dimensions

We are given the radius of the cone's base \(r = 7\text{ in}\) and the slant height \(y = 25\text{ in}\).

Calculate the height of the cone

The radius \(r\), height \(x\), and slant height \(y\) form a right triangle. We use the Pythagorean theorem to find \(x\):
\[
x^2 + r^2 = y^2
\]
\[
x^2 + 7^2 = 25^2
\]
\[
x^2 + 49 = 625
\]
\[
x^2 = 576
\]
\[
x = 24\text{ in}
\]

Calculate the volume of the cone

The formula for the volume \(V\) of a cone is:
\[
V = \frac{1}{3}\pi r^2 h
\]
Substitute \(r = 7\) and \(h = x = 24\):
\[
V = \frac{1}{3}\pi (7)^2 (24)
\]
\[
V = \frac{1}{3}\pi (49)(24)
\]
\[
V = 8 \cdot 49\pi
\]
\[
V = 392\pi\text{ in}^3
\]
</reasoning>

<answer>
<mcq-correct>(A) \(392\pi\text{ in}^3\)</mcq-correct>
<mcq-option>(B) \(408\pi\text{ in}^3\)</mcq-option>
<mcq-option>(C) \(112\pi\text{ in}^3\)</mcq-option>
<mcq-option>(D) \(117\pi\text{ in}^3\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify the given dimensions

We are given the radius of the cone's base \(r = 7\text{ in}\) and the slant height \(y = 25\text{ in}\).

Calculate the height of the cone

The radius \(r\), height \(x\), and slant height \(y\) form a right triangle. We use the Pythagorean theorem to find \(x\):
\[
x^2 + r^2 = y^2
\]
\[
x^2 + 7^2 = 25^2
\]
\[
x^2 + 49 = 625
\]
\[
x^2 = 576
\]
\[
x = 24\text{ in}
\]

Calculate the volume of the cone

The formula for the volume \(V\) of a cone is:
\[
V = \frac{1}{3}\pi r^2 h
\]
Substitute \(r = 7\) and \(h = x = 24\):
\[
V = \frac{1}{3}\pi (7)^2 (24)
\]
\[
V = \frac{1}{3}\pi (49)(24)
\]
\[
V = 8 \cdot 49\pi
\]
\[
V = 392\pi\text{ in}^3
\]
</reasoning>

<answer>
<mcq-correct>(A) \(392\pi\text{ in}^3\)</mcq-correct>
<mcq-option>(B) \(408\pi\text{ in}^3\)</mcq-option>
<mcq-option>(C) \(112\pi\text{ in}^3\)</mcq-option>
<mcq-option>(D) \(117\pi\text{ in}^3\)</mcq-option>
</answer>

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