Question

the radius of the cone is 7 in and y = 25 in. what is the volume of the cone in terms of π? 117π in³ 112π in³ 408π in³ 392π in³

Explanation:

Step1: Recall the volume formula for a cone

The volume formula of a cone is $V=\frac{1}{3}\pi r^{2}h$. First, we need to find the height of the cone. Using the Pythagorean - theorem in the right - triangle formed by the radius ($r$), height ($h$), and slant height ($y$) of the cone. The Pythagorean theorem is $y^{2}=r^{2}+h^{2}$, where $r = 7$ in and $y = 25$ in.

Step2: Calculate the height of the cone

We can solve for $h$ (the height of the cone) from $y^{2}=r^{2}+h^{2}$. Rearranging the formula gives $h=\sqrt{y^{2}-r^{2}}$. Substitute $r = 7$ and $y = 25$ into the formula: $h=\sqrt{25^{2}-7^{2}}=\sqrt{(25 + 7)(25 - 7)}=\sqrt{32\times18}=\sqrt{576}=24$ in.

Step3: Calculate the volume of the cone

Now that we have $r = 7$ in and $h = 24$ in, we substitute these values into the volume formula $V=\frac{1}{3}\pi r^{2}h$. So $V=\frac{1}{3}\pi\times7^{2}\times24$. First, calculate $7^{2}=49$ and $\frac{1}{3}\times24 = 8$. Then $V=8\times49\pi=392\pi$ in³.

Answer:

$392\pi$ in³