Part b please 4. A cylindrical water tank has a height of 8 meters and radius 2 meters.If the tank is full of water, how much work is required to pump the water to the level ofthe top of the tank and out of the tank?b. Is it true (or false) that it takes half as much work to pump the water out of the tanka.when it is half full as when it is full? (The answer to this part must include a solvedintegral representing work and at least one complete sentence as your answer.)

Answered: 4. A cylindrical water tank has a…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Part b please

4. A cylindrical water tank has a height of 8 meters and radius 2 meters.
If the tank is full of water, how much work is required to pump the water to the level of
the top of the tank and out of the tank?
b. Is it true (or false) that it takes half as much work to pump the water out of the tank
a.
when it is half full as when it is full? (The answer to this part must include a solved
integral representing work and at least one complete sentence as your answer.)

Expert Solution

Step 1

No ,it isn't true It's false . Because work done does not dependence only upon $h$ (as in Work done to empty full filled tank $= \frac{π}{2} ρ g r^{2} h^{2}$ proved in part $(i)$ ), it is proportional to $h^{2} .$

Now, We know that work done to pump this water $= π ρ g {(r)}^{2} y d y$

Work done to empty half full tank

$= \int_{\frac{h}{2}}^{h} πρg (r^{2}) y d y$ ( integrate from $y = \frac{h}{2} t o y = h$ )

$= {πρgr}^{2} {(\frac{y^{2}}{2})}_{h / 2}^{h} = \frac{1}{2} {πρgr}^{2} [{(h)}^{2} - {(\frac{h}{2})}^{2}] = \frac{1}{2} {πρgr}^{2} \times \frac{3 h^{2}}{4}$

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