If the height and diameter of the cylinder are halved, by what factor will the volume of the cylinder change?

volume of a cylinder is V = πr²h, where r = radius and h = height.

now, if you cut the diameter by half, you also cut the radius by half, so we'd end up with r/2 instead.

if you cut the height in half, we'd end up with h/2.

then,

[tex]\bf \textit{volume of a cylinder}\\\\ V=\pi r^2 h\quad \begin{cases} r=\frac{r}{2}\\\\ h=\frac{h}{2} \end{cases}\implies V=\pi \left( \frac{r}{2} \right)^2\left( \frac{h}{2} \right)\implies V=\pi \left(\frac{r^2}{2^2} \right)\frac{h}{2} \\\\\\ V=\pi \cdot \cfrac{r^2}{4}\cdot \cfrac{1}{2}\cdot h\implies V=\cfrac{1}{4}\cdot \cfrac{1}{2}\cdot \pi r^2 h\implies V=\cfrac{1}{8}(\pi r^2 h)[/tex]

notice, the new size is just 1/8 of the original size.