Question

As a hot-air balloon rises vertically, its angle of elevation from a point P on level ground d = 140 kilometers from the point Q directly underneath the balloon changes from 15°10' to 29°30' (see the figure). Approximately how far does the balloon rise during this period? (Give the answer to one decimal place.)
For height at lower angle:
tan 15°10' = height1/140
height1 = 140tan15°10'

in the same way:
height2 = 140tan29°30'

rise of the balloon in that change of angle
= 140tan29°30' - 140tan15°10'
= 41.256

I am sure your units are not correct.
How can they even see the balloon from 140 km away ?

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