why does the height of the cumulus cloud base change
Due to weather factors such as instability, moisture, and temperature gradient
Cumulus cloud
Cumulus clouds are often precursors of other types of clouds, such as cumulonimbus, when influenced by weather factors such as instability, moisture, and temperature gradient. 
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Cumulus cloud [SEP] Introduction
Cumulus clouds are clouds which have flat bases and are often described as "puffy", "cotton-like" or "fluffy" in appearance. Their name derives from the Latin "cumulo-", meaning "heap" or "pile". Cumulus clouds are low-level clouds, generally less than in altitude unless they are the more vertical cumulus congestus form. Cumulus clouds may appear by themselves, in lines, or in clusters. Cumulus clouds are often precursors of other types of clouds, such as cumulonimbus, when influenced by weather factors such as instability, moisture, and temperature gradient. Normally, cumulus clouds produce little or no precipitation, but they can grow into the precipitation-bearing congestus or cumulonimbus clouds.