A company proposed to build a water detention tank to address the flooding problem in a village. To

Question

A company proposed to build a water detention tank to address the flooding problem in a village. To test the feasibility of the proposal, the company created a model of a water detention tank, with a capacity of $300$ cm$ {}^3 $.

Water is pumped into the model of the tank at a constant rate of $50$ cm$ {}^3 $/min and pumped out at a rate proportional to the volume of water in the tank. At time $t$ minutes, the volume of water in the model of the tank is $V$ cm$ {}^3 $.

Write down a differential equation for this situation.[1]
Solve this differential equation to obtain the general solution for $V$ in terms of $t$.[4]

Initially, the volume of water in the tank is $100$ cm$ {}^3 $. After a minute, the volume increased to $130$ cm$ {}^3 $.

Hence, show that the particular solution for $V$ in terms of $t$ is $V = 289 - 189e^{-0.173t}$, correct to $3$ significant figures.[4]
Find how long it will take for the tank to reach $95$% of its maximum capacity.[1]
Sketch the graph of $V$ against $t$.[2]
Explain, with justification, what will happen to the volume of water in the tank in the long run.[1]

Accepted Answer

(i) $\frac{\mathrm{d}V}{\mathrm{d}t}=50-kV$, $k>0$ (ii) $V=\frac{50}{k}+Ce^{-kt}$ (iii) $V=289-189e^{-0.173t}$ (iv) $22.9\text{ minutes}$ (v) Increasing concave-down curve from $(0,100)$ with horizontal asymptote $V\approx289$ (vi) $V\to289\text{ cm}^3$, so the tank does not overflow.

2024 TMJC P1 Q10

2024 TMJC P1 Q10

Video Solution:

Solution:

Solution locked