A bucket has the shape of a frustum of a cone, with a circular bottom of radius $2$ and an open circular top of radius $5$. Starting when the bucket is empty, a hose that runs water at a constant rate takes $38$ minutes to fill the bucket up to one third of its height. How many more minutes will it take to fill the rest of the bucket?

1. Extend the lateral surface to the apex $V$ of the cone. Every horizontal slice is a circle, and its radius is a linear function of the height, so the cones cut off at the apex are all similar.
2. At one third of the height the radius is $2+\frac13(5-2)=3$. The cones with apex $V$ and base radii $2$, $3$, $5$ have volumes proportional to $2^3,3^3,5^3$, i.e. to $8$, $27$, $125$.
3. The water up to one third of the height has volume proportional to $27-8=19$ and takes $38$ minutes, so each unit takes $2$ minutes. The remaining part has volume proportional to $125-27=98$, so it takes $98\cdot2=\boxed{196}$ more minutes.