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let us consider following problem:

When $30$ gallons water poured in to cylinder whose sides are perpendicular to base, water level rises $0.5$ ft, if $7.5$ gallons occupy $1$cu.ft space what is the area of base of tank?

so as i understand we have cylinder,which's volume is equal to

$\pi*r^2*h$

now when we have poured $30$ gallon,then water level rises $0.5$ ft,what does it means?does it means that it's volume is increased by $0.5$ or height?also what i think if $7.5$ gallon occupy volume of $1$cu.ft,which means that $\pi*r^2*h=1$,then how can i connect these two information together?can i say that $30$ gallon occupy $4$ cu.ft,because $30/7.5=4$,so it means that

$\pi*r^2*h=1$

$\pi*r^2*(h+0.5)=4$? if it is so then i will got $\pi*r^2*0.5=3$ from which $\pi*r^2=6$ is it correct?

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2 Answers

up vote 1 down vote accepted

Not quite. $30$ gallons take up $4$ cubic feet of space, as you've determined, but the rest doesn't really make sense. You seem to be starting with one cubic foot of water in the tank, so when we pour in $30$ gallons ($4$ cubic feet), we have a change in water level of $0.5$ feet, yes, but our new volume of water is $5$ gallons. That is, if $$\pi r^2h=1,$$ then $$\pi r^2(h+0.5)=5,$$ whence $\pi r^2\cdot0.5=4,$ and you can solve from there.

More simply, we could just assume that we started with an empty tank and poured in $30$ gallons ($4$ cubic feet), so that our change in water level is exactly the water level. That is, $$\pi r^2\cdot0.5=4,$$ as before.

Edit: Instead of making such an assumption, let us instead assume that $V$ is the volume of water originally in the tank and $h$ is the initial water level (in feet). The cylindrical shape tells us that $$\pi r^2h=V,$$ where $r$ is the radius of the tank (in feet). The addition of $4$ cubic feet of water increases the water level by $0.5$ feet, meaning that $$\pi r^2(h+0.5)=V+4.$$ Once again, we find that $$\pi r^2\cdot0.5=4.$$

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but why $5$?because $7.5$ takes $1$,$30$ takes $4$ right? –  giorgi Aug 9 '13 at 21:23
    
Your original equation, $$\pi r^2h=1,$$ indicates that we have $1$ gallon in the tank, and the water depth $h$ is whatever the appropriate depth is for that volume. In order to increase the height by $0.5$ feet--which is what the $h+0.5$ in your second equation indicates--we must pour an additional $4$ cubic feet into the tank with the water that was already in the tank, bringing the total volume to $5$ gallons, not just $4$. –  Cameron Buie Aug 9 '13 at 21:29
    
so statement that if 7.5 gallons occupy 1cu.ft space means.. –  giorgi Aug 9 '13 at 21:31
    
It simply allows you to convert $30$ gallons into its equivalent volume in cubic feet. –  Cameron Buie Aug 9 '13 at 21:32
    
what i was thinking is that,if 7.5 occupy 1 cu.feet,then 30 means already 7.5+22.5,which means $3*7.5$,so 1+3 –  giorgi Aug 9 '13 at 21:33
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The $\pi r^2h=1$ is not true. They mean that you can use the $30/7.5=4$ as you described such that: $$\pi r^2(.5)=4$$

I think you understand this well enough to do the rest. The initial height of water in the tank does not matter.

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you mean that area of base is $8$? –  giorgi Aug 9 '13 at 21:20
    
yes that is the result –  kaine Aug 9 '13 at 21:24
    
ok but why is not first equation true? i meant $\pi*r^2*h=1$ –  giorgi Aug 9 '13 at 21:26
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