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A cylindrical tank is 4 feet high and has on outer diameter of 2 feet. The walls of the tank are 0.2 inches thick. We need to approximate the volume of the interior of the tank assuming that the tank has a top and a bottom that are both also 0.2 inches thick.

We approximate a function in the following manner :

$\Delta f=f_{x}\Delta x+f_{y}\Delta y$ , where $f=f(x,y)$.

So we let $V=\Pi R^{2}H$ ,$V_R=2\Pi R^{}H$ and $V_H=\Pi R^{2}$ ,

Here , $\Delta R$ would be $\dfrac{0.2}{12}$ feet , and $\Delta H$ would be $\dfrac{2*0.2}{12}$ feet .(For top and bottom)

Thus , we write : $\Delta V = (2\Pi RH)\dfrac{0.2}{12}+ (\Pi R^{2})\dfrac{2*0.2}{12}$ , where we put $R=1$ and $H=4$.

Am I correct for this approximation ?

User9523
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  • Why do you divide by 12 ? – callculus42 Aug 22 '15 at 15:02
  • Height and diameter are given in "feet" , thickness is given in "inches"..@calculus Sorry , just made an edit.. – User9523 Aug 22 '15 at 15:06
  • Ah ok. It looks fine. Maybe you should take the negative value, because the change is negative. – callculus42 Aug 22 '15 at 15:13
  • Change is negative ? How ? I am sorry , i am not getting it .. Would you care to elaborate please ?@calculus – User9523 Aug 22 '15 at 15:16
  • You have the volume including the walls. Then you approximate the volume without the walls, which is smaller than the volume including the walls. Therefore the change is negative. It is comprehensible ? – callculus42 Aug 22 '15 at 15:25
  • I got your point .. The "change" should be negative.. But I have another doubt.. $\Delta V$ gives the change in the volume , but we need the volume of the inside of the tank.. So , shouldn't it be $V- \Delta V$ ? @calculus – User9523 Aug 22 '15 at 16:25
  • Yes. To be straigtforward it is $\Delta R=-\dfrac{0.2}{12}$ and $\Delta H=-\dfrac{2 \cdot 0.2 }{12}$. By using the negative changes of R and H the result has a negative sign without any other manipulations. – callculus42 Aug 22 '15 at 16:34
  • Yes. You are right. The inner volume is $V^I=V^0-\Delta V$. – callculus42 Aug 22 '15 at 16:59

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