Chapter 2.3, Problem 9P

Imagine a medieval world. In this world a Queen wants to poison a King, who has a wine keg with $500\text{L}$ of his favourite wine. The Queen gives a conspirator a liquid containing $5g/\text{L}$ of poison, which must be poured slowly into the keg at a rate of $0.5\text{L}/\min$. The poisoner must also remove the well-stirred mixture at the same rate, so that the keg is not suspiciously full.

Find a formula for the amount of poison in the keg at any time, measured from the start of the pouring by the poisoner.

A plot is hatched for the King to drink wine from the keg while he is on a hunt, where he will become so addled that his prey will surely kill him. The poisoner must pour for a time $T$, when the poison in the keg reaches a dangerous concentration of $0.005g/L.$ Find $T$.

The Lord High Inquisitor of the Realm never learned about differential equations. Nonetheless, knowing the basic numbers (keg size, poison concentration, etc.), he can produce an estimate for the time $T$ that the poisoner was at the keg. In fact, his estimate is within $2\%$ of the exact value found in part $(b)$. What is the Lord’s estimate? In the context of differential equations, why is it so close to the exact value obtained from the solution?