FAC1004 Tutorial 14 — Mixing Tank Differential Equations

Centre for Foundation Studies in Science, University of Malaya FAC1004 Advanced Mathematics II, Session 2025/2026

Problem 1: Cylindrical Tank with Brine Solution

A cylindrical tank holds $1500$ liters of brine solution with $90$ g of salt. Pure water is pumped in at the rate of $30$ liters/second. The solution is thoroughly mixed and then pumped out at the rate of $30$ liters/second. Find:

(a) The differential equation for the above situation. Hence, find amount of salt in the tank after $t$ seconds.

(b) The amount of salt in the tank after $25$ seconds.

(c) The time for the amount of salt in the remaining solution to be $35$ g.

Problem 2: Chlorinated Container

A $300$ ft$^3$ container is chlorinated with $15$ m$^3$ of chlorine. A $10$ m$^3$/ft$^3$ mixed solution flows into the container at the rate of $16$ ft$^3$/minute. If the mixed solution flows out from the container at the same rate, find:

(a) The differential equation for the scenario given above. Hence, find amount of chlorine in the container after $t$ minutes.

(b) The amount of chlorine in the container after $60$ seconds.

Problem 3: Sugar Bowl (Variable Volume)

A $400$ m$^3$ bowl contains $120$ m$^3$ distilled water mixed with $595$ g sugar. Solution A with $5$ g/m$^3$ sugar concentration is pumped into the bowl at the rate of $3$ m$^3$/second. Assume the solution is uniformly mixed, calculate the amount of sugar in the bowl after $2$ minutes if the solution flows out of the bowl at the rate of $2$ m$^3$/second. Hence, find amount of the sugar when the solution overflows.

Problem 4: Variable Outflow Variations

Based on Question 3, calculate the amount of sugar in the bowl, if the solution flows out at the rate of:

(a) $1$ m$^3$/second after $2$ minutes. (b) $6$ m$^3$/second after $20$ seconds.

Hence, determine the amount of the sugar when the tank overflows/empties.

Problem 5: Mercury-194 Decay

A Mercury-194 decays exponentially. The decay rate is $72.8\%$ per hour and can be modeled by

$$\frac{dP}{dt} = -0.728P$$

(a) Find an equation that satisfies the above differential equation. Assume $P(0) = P_0$.

(b) If the initial amount of Mercury-194 is $650$ g, what is the amount of Mercury-194 left after $50$ minutes?

(c) What is the half-life of the Mercury-194?

Problem 6: Plutonium-241 Decay

A Plutonium-241 decays at a rate proportional to the amount present. At $t = 0$, the amount of Plutonium-241 is $400$ mg and after two years, $15\%$ has decayed.

(a) Find an equation to model this above situation. (b) How much Plutonium-241 left after $6$ years? (c) What is the half-life of Plutonium-241?

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