jmc

We start by finding the volume of the bucket and the volume of the barrel. Let $r$ be ten inches. The bucket is half of a sphere of radius $r$, so the volume it can hold is $$\frac{1}{2}\cdot \frac{4}{3}\pi r^3=\frac{2}{3}\pi r^3.$$On the other hand, the barrel is a cylinder of radius $r$ and height $15\text{ in}=\frac{3}{2}r$, so its volume is $$\pi r^2\cdot \text{height}=\frac{3}{2}\pi r^3.$$Therefore the ratio of the volume of the barrel to the volume of the bucket is $$\frac{(3/2)\pi r^3}{(2/3)\pi r^3}=\frac{9}{4}=2+\frac{1}{4}.$$Thus two buckets will not suffice to fill the barrel, but three will, so Brad needs $\boxed{3}$ trips to the well.