The three forces acting on a hot-air balloon that is moving vertically are its weight, the force due to air resistance and the upthrust force. the hot-air balloon descends vertically at constant speed. the force of air resistance on the balloon is f. which weight of material must be released from the balloon so that it ascends vertically at the same constant speed?

Let
upthrust = T
weight = W = mg
Air resistance = F

When balloon is descending, air resistance acts upwards (positive)
By Newton's first law, the net force on the balloon is zero, or
T+F-W=0(1)

Let w=weight of material dumped so that balloon now travels upwards at constant speed.
Air resistance acts against motion, namely downwards.
The Newton's equation now reads
T-F-(W-w)=0(2)

Subtract (2) from (1)
T+F-W - (T-F-(W-w)) = 0
Solve for w
w=2F, or
the WEIGHT of material to be released equals twice the resistance of air.