Question 1146036
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            To the author of this problem / post.


            There is a FATAL discrepancy / inconsistency in terminology you use.



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When you describe function P(t), you give its dimension unit as "gallons per minute".


Such a function is (has the technical name) "rate of inflow".

But if it is "rate of inflow", it is NOT the volume of the water in the tank.



The same thing is with the function L(t).



When you describe the function L(t), you give its dimension unit as "gallons per minute".


Such a function is (has the technical name) "rate of emptying", or "rate of drainage", or "rate of outflow".

But if it is "rate of emptying", it is NOT the volume of the water in the tank.



Using correct terminology is a MUST when you formulate a problem.


Therefore, there is no any sense in solving the problem as it is presented in the post, until it is fixed.


Make all necessary corrections, and then re-post the problem to the forum.


Do not post it to me personally.
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By the way, when you re-think the problem to the end, you may find that it EITHER has no sense at all,


OR has totally different meaning than you thought before.