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:

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 \(h = x = 24\):

$$ V = \frac{1}{3}\pi (7)^2 (24) $$

$$ V = \frac{1}{3}\pi (49)(24) $$

$$ V = 8 \cdot 49\pi $$

$$ V = 392\pi\text{ in}^3 $$

Answer:

• (A) \(392\pi\text{ in}^3\) (Correct answer)
• (B) \(408\pi\text{ in}^3\)
• (C) \(112\pi\text{ in}^3\)
• (D) \(117\pi\text{ in}^3\)