Why does a hot-air balloon rise? the weight of the balloon is less than the weight of the air displaced by the balloon. the weight of the air displaced by the balloon is less than the volume of the balloon. the volume of the air displaced by the balloon is less than the volume of the balloon.

The weight of the air displaced by the balloon is less than the volume of the balloon.

Explanation:

A hot air balloon is a cloth wrap that contains several thousand cubic meters of air inside (a large volume of air). The burner heats the liquid propane to a gaseous state to generate a huge flame, which can reach more than 3 meters, thus heating the air mass inside the balloon. In this way, its density is modified with respect to the air that surrounds it, because the hot air is lighter than the outside air (less dense), causing the balloon to rise and float.

Now, if we know that the density of a body is directly proportional to its mass and inversely proportional to its volume :

We can deduce that by increasing the volume of the body, its density will decrease.

This is proof of Archimedes' Principle:  

A body totally or partially immersed in a fluid at rest, experiences a vertical upward thrust equal to the mass weight of the body volume that is displaced.

In this case the fluid is the air outside. So, the warm air inside the balloon, being less dense, will weigh less than the outside air and therefore will receive an upward pushing force or thrust that will make the balloon ascend.