C.29.BB:5.1 - A transfer line changes which balance answers the tank question

Use a constant-density liquid, measured by volume in litres. At the start, tank A holds 30 L and tank B holds 10 L. Over the chosen interval, an external supply adds 12 L to A, and an external outlet removes 7 L from B.

First suppose the line transfers 15 L from A to B with no change in its liquid content. Then

A: 30 + 12 - 15 = 27 L
B: 10 + 15 - 7 = 18 L
A+B: 40 + 12 - 7 = 45 L.

The internal 15 L cancels. The larger balance gives the combined amount without needing that internal transfer, while the separate balances give each tank’s amount.

Now change the condition: the line begins empty, receives 15 L from A, delivers 13 L to B, and retains 2 L. The results become

A: 27 L
B: 10 + 13 - 7 = 16 L
line: 0 + 15 - 13 = 2 L.

The two tanks contain 43 L. Their boundary excludes the line, so 15 L leaves that boundary and 13 L re-enters. The combined tank change is 12 - 7 - 15 + 13 = 3 L. Including the line gives 45 L and recovers the five-litre increase from the external exchanges alone.

If the line’s increase is known only to lie between 0 and 2 L, the final tank total lies between 43 and 45 L. This bound settles a request for at least 42 L in the tanks. A request for at least 44 L needs a tighter bound or the line’s actual content. No additional observation is required for the first request.