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I'm stuck at this problem and the solution doesn't make sense to me, I would like to understand what's wrong with my reasoning.

There is a cylindrical tank with height 15m and radius 10m, fluid is being pumped into a biddon, which is composed of a rectangle and a prism (see picture).

Cylindrical and prismatic reservoir

We know that the liquid level is decreasing with 2cm/sec in te cylindrical tank, and want the rate the level increases in the biddon at a level of 6m

My way of solving is first calculate $V^\prime$.

$V=hr^2\pi$ so $V^\prime=\pi r^2h^\prime\, \to\, V^\prime=10000(2\pi)=20000\pi$

Then for the body I split up in two parts, First $V=1000(1500h)$

Second $V=(800\cdot1500h)/2$

(I did $\frac{30}{10}=\frac{x}{6}\,\to\,x=18$ and $b=18-10$)

Added them together $V=15\cdot10^5h+6\cdot10^5h \,\to\, 21\cdot10^5h$

Then $\frac{V^\prime}{21\cdot10^5}=h^\prime$ so $h^\prime=0.01496\,$cm/sec

However the result should be 0.019 cm/s, and it involves the height of the cylinder, but I don't see how that matters, or why my way is wrong.

I would really appreciate your help :)

1 Answers 1

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When the depth of water in the biddon is $y$ meters the volume $V_B$ will be \begin{eqnarray} V_B&=&150y+\frac{1}{2}\cdot(15)(y)(2y)\\ &=&15y^2+150y \end{eqnarray}

Therefore

\begin{equation} \frac{d}{dt}V_B=30(y+5)\frac{dy}{dt}\tag{1} \end{equation}

The water level in the cylinder is dropping at a rate of $0.02$ m/s which means the volume $V_C$ of the water in the cylinder is decreasing at a constant rate of $\frac{dV_C}{dt}=-\pi(10)^2(0.02)=-2\pi\,$m$^3$/sec.

But this means that $V_B$ is increasing at a constant rate $\frac{dV_B}{dt}=2\pi\,$m$^3$/sec.

So to find the rate $\frac{dy}{dt}$ at which the water level is rising in the biddon when $y=6$ we substitute into equation $(1)$ to get

\begin{equation} 2\pi=330\,\frac{dy}{dt} \end{equation}

and get the solution

\begin{equation} \frac{dy}{dt}=\frac{\pi}{165} \end{equation}

So when $y=6$ m, the water level is rising approximately $0.019$ m/sec or $1.9$ cm/sec.

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    Thank you! I didn't realise that the prysm had a length of twice the hight, it all makes sense now – 2017-01-22
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