For similar 3D shapes, the ratio of volumes is not the same as the ratio of heights. How does volume change when every length is multiplied by a factor of $k$?

Once you know the volume scale factor from smaller to larger, multiply the smaller cone’s volume, $27\pi$, by this factor. Be careful to use $k^3$, not just $k$ or $k^2$.

After you decide on the correct scale factor for volume, multiply that number by 27 to get the new coefficient in front of $\pi$. Double-check the multiplication to avoid small mistakes.

In Desmos, type 27 * pi * 4^3 to represent the smaller cone’s volume times the cube of the linear scale factor from smaller to larger.

Look at the numeric result Desmos shows for 27 * pi * 4^3. Identify the integer coefficient in front of $\pi$; that coefficient, with $\pi$ attached, gives the volume of the larger cone.

For similar 3D shapes (like cones), all linear dimensions (height, radius, slant height) are multiplied by the same scale factor. Here, the height of the larger cone is 4 times the height of the smaller cone, so the linear scale factor from smaller to larger is $4$.

Volume is a 3D measure, so when all linear dimensions are multiplied by a factor $k$, the volume is multiplied by $k^3$.

Here, $k = 4$, so the volume scale factor from the smaller cone to the larger cone is

The smaller cone’s volume is given as $27\pi$ cubic centimeters.

To get the larger cone’s volume, multiply the smaller volume by the volume scale factor $64$:

Now compute $27 \times 64$ to find the final volume.

Multiply:

Alternatively:

So the larger cone’s volume is

Therefore, the correct answer is $1{,}728\pi$.