A cylinder holds 100 millimeters. If the radius of the base is doubled and the height is halved, new volume?

It would be doubled or 200 mm.

The formula for volume of a cylinder is V=(1/2)bh where the base is pi*r^2 so V=(1/2)(pi*r^2)h

So by doubling the radius and halving the height you would get

V=(1/2)(pi*(2r)^2)(h/2)=(1/2)(pi*4r^2)(h/2)=pi*r^2*h

You notice how the half disappeared from the equation showing that you need to double the initial volume to equal the new equation. This occurs because the radius is a square ratio while the height is a linear, so by increasing the radius you do so by a square factor and vice versa.

Let's start with a cylinder with a radius of 2 and a height of 7.955. It holds 100 mm.

Now let's increase the radius to 4 and the decrease the height to 3.9775. It will hold 200 mm. The formula for volume of a cylinder is pi * radius² * height.