Example 3-6-5 - Maple Help

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Chapter 3: Applications of Differentiation



Section 3.6: Related Rates



Example 3.6.5



A right-circular conical tank, whose cross-section through its axis is shown in Figure 3.6.4, is being filled with water at the constant rate $\mathrm{λ}$.

At time $\stackrel{^}{t}$, find $\stackrel{.}{h}\left(\stackrel{^}{t}\right)$, the rate of change of the height of the water, where $\stackrel{^}{t}$ is the moment when the volume is $k$ times the volume of the tank, $0.

The dimensions of the tank and the rate of fill are all in consistent units. The height of the tank is $H$, while the radius of the opening is $R$. The varying radius of the circle at the level of the water is $r\left(t\right)$ (green dotted line in Figure 3.6.5(a)), and the varying height of the water is $h\left(t\right)$.

Hint: The volume of the tank is

 > p1:=plot([[0,0],[2,5],[-2,5],[0,0]],style=line,color=black): p2:=plot([[0,0],[0,5]],style=line,linestyle=dot,color=red): p3:=plot([[0,3.5],[7/5,3.5]],style=line,linestyle=dot,color=green): p4:=plots:-textplot({[1,5.2,typeset(R)],[-1,5.2,typeset(R)],[-.3,4.3,typeset(H)],[.7,3.3,typeset(r(t))],[.3,2.4,typeset(h(t))]},font=[Times,12]): p5:=plots:-textplot({[-.2,3.5,A],[1.6,3.5,B],[.2,0,O],[0,5.2,C],[2,5.2,E]},font=[Times,BoldRoman,14]): plots:-display(p||(1..5),scaling=constrained, axes=none);

 >

Figure 3.6.5(a)   Conical tank







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