Final answer:
The sphere, having the lowest surface area to volume ratio for a given volume, would make the best riser as it solidifies more slowly, allowing for efficient feeding of the casting.
Step-by-step explanation:
When considering the Total Solidification Times (TST) of three casting shapes and determining which would make the best riser, we must consider the surface area to volume ratio of the shape. A sphere, when keeping the volume constant, presents the minimum surface area compared to other shapes, which would result in a slower solidification time. This is beneficial for a riser as it allows more time for the liquid material within a mold to feed the cast as it solidifies. On the other hand, shapes like filaments and sheets, with their greater surface area to volume ratio, solidify more rapidly. Therefore, based on the knowledge given, the sphere would make the best riser because of its compact shape which results in a lower surface area to volume ratio, thus slower cooling and solidification.