# Work compute with integral

• May 12th 2008, 03:51 AM
RedBarchetta
Work compute with integral
A tank is full of water. Find the work W required to pump the water out of the spout. Use the fact that water weighs 62.5 lb/ft3.
a = 5 b = 9 c = 12
http://www.webassign.net/scalcet/6-4-024alt.gif

So for this one, you have to set up an amount of rectangles.

The biggest rectangle is 5(or y) by 9, in relation to a smaller rectangle that is w by 9. I can also setup a relationship between the two to find that y/w=9/9 or y=w. So therefore we can produce an equation for the area of one rectangle.

dA=9y

Now the thickness of this rectangle is extremely small so the thickness is dy.

dV=9y dy

The density of water is 62.5 lb/ft^3, so this one rectangular-box of water is:

dM=562.5y dy

To find the force, we multiply by the force of gravity, I used 32.2 ft/sec^2.

dF=18112.5y dy

So our work is dF times the distance. This is where I become lost. Would the distance be vertical or horizontal? would it be (5-y) or (12-y)?
• May 12th 2008, 07:35 PM
RedBarchetta
Well, after working the problem I came up with 50625 ft-lb. Wrong. Ugh.

I tried using a linear relationship. y=12/5y+12

dV=9(12/5y+12) dy
dV=108/5*y + 108

I multiplied by the density of water(62.5 lb/ft^3). d=m/v then d*v=m.

dV=1350y+6750 dy

Now I integrate from zero to five.

[675x^2+6750x](zero to five)=50625 ft-lb...
• May 13th 2008, 05:53 PM
RedBarchetta
Anyone?

Thank you.
• May 13th 2008, 06:58 PM
RedBarchetta
^^ I just entered your answer into my homework checker and it told me I was wrong.
• May 14th 2008, 04:29 AM
galactus
I am sorry, I didn't notice the spout. Let me get back. I am at work now and busy.
• May 14th 2008, 07:30 AM
galactus
I think what we forgot to do was multiply by x.

By similar triangles, $\frac{5}{12}=\frac{5-x}{y}, \;\ y=\frac{12}{5}(5-x)$

$(62.5)(12/5)(9)=1350$

$1350\int_{0}^{5}x(5-x)dx=28125 \;\ ft/lbs$
• May 14th 2008, 01:00 PM
RedBarchetta
Yes! Finally, with one try left I get it. Thanks Galactus. It's quite the problem.
• May 14th 2008, 01:13 PM
galactus
That was correct then?. I thought so. I just forgot the x last time.