Final answer:
The ratio of the volumes of two cones with heights in the ratio of 1:3 and radii in the ratio of 3:1 is 3:1.
Step-by-step explanation:
To determine the ratio of the volumes of two cones with differing heights and radii, we must first understand the formula for the volume of a cone, which is given by V = (1/3)πr²h, where V is volume, r is the radius, and h is the height of the cone.
If the heights of the cones are in the ratio 1:3, and the radii are in the ratio 3:1, we can represent their respective volumes as V1 = (1/3)π(3r)²(1h) and V2 = (1/3)π(r)²(3h), where V1 is the volume of the first cone and V2 is the volume of the second cone. Simplifying these expressions, we get V1 = 9πrh/3 and V2 = 3πrh.
Finally, the ratio of volumes V1 to V2 is 9/3, which simplifies to 3:1.