Work and Fluid Force, lifting water out of a triangular prism tank

schlynn

1. The problem statement, all variables and given/known data
A vertical cross section of a tank is shown. Assume the tank is 16 feet long and full of water. ([tex]\delta=62.4[/tex], and that the water is to be pumped to a height of 8 feet above the top of the tank. Find the work done in emptying the tank. The tank is a triangular prisim with base=5ft and height=8ft.


2. Relevant equations
Not sure that there are any.


3. The attempt at a solution
First, I am supposed to find the force of the water. It says to find the width as a function of the height, and the book is unclear how to do this very well. From what I can gather it's the height divided by half the base equals the distance to pump the water divided by W, W being the width. So I solved and got W=[tex]\frac{5}{2}-\frac{5}{16}y[/tex]. And then since work is force times height, you just multiply that by the distance it has to be lifted, that's 16-y. And that's your integrad, but you have a 16[tex]\delta[/tex] factor too, but you just bring that outside of your integral. After I simply the integrand and anti differentiate I got [tex]40y-\frac{15}{4}y^{2}+\frac{5}{48}y^{3}[/tex] evaluated from 0 to 8, again, with the factor of 16[tex]\delta[/tex]. Fundamental theorm it and I got 132787 rounded to the nearest whole number. The answer is wrong, and I'm fairly certain I know how to do all of this except finding the width as a function of the height. The book says it has to do with similar triangles. But I don't get what they are saying. Can someone shed some light on this for me?

lanedance

how about using conservation of energy and consider the centre of mass's

schlynn

Because this is a calculus 2 class, not a physics class, I don't know how to do it that way. I know how to find the centroid of an area that has uniform density, but that's not how we are supposed to do it.

lanedance

ok well i would find w(h)

then infinetsimal vol element
dV = w(h).L.dh

think of the work dW required to get this infinitesimal element to teh reuired hieght and then integrate over h

its all the same thing though

schlynn

That's what I'm having trouble with. I can't find the infitesimal volume, I can't get the width as a function of the height. Could you walk me through it?

lanedance

changed notation in last post

ok so you know
w(0) = 5
w(8) = 0

and as its a triangle, its width will vary linearly in between...

so basically you have two points (0,5) and (8,0) find the equation of the line that connects them, and that will be w(h)