Calculating Work to Empty a Water-Filled Trough

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In summary, the problem involves finding the amount of work in foot-pounds required to empty a trough filled with water. The trough is 2 feet long and 1 foot high, with a cross-section shaped like the graph of x^6 from x=-1 to x=1. The volume of the trough can be calculated by taking the integral of the area of the cross-section with respect to y. The area is a rectangle with a length of 2 and a width of the distance from -x to x, where x^6=y. The weight of the water is determined by multiplying the volume by the density of water, which is 62 pounds per cubic foot. The distance traveled is 1 - y. Therefore, the
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chimbooze
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A trough is 2 feet long and 1 foot high. The vertical cross-section of the trough parallel to an end is shaped like the graph of x^6 from x=-1 to x=1. The trough is full of water. Find the amount of work in foot-pounds required to empty the trough by pumping the water over the top. Note: The density of water is 62 pounds per cubic foot.

Work = weight x distance traveled.

Weight = volume x density
Weight =

distance traveled = 1 - y

I'm just having issue with coming up with the volume for the cross section of the trough. If I know that, I can take the integral from 0 to 1 with respect to y.
 
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The volume is the area of the cross section*dy. The area is a rectangle. The length is 2 and the width is the distance from -x to x, where x^6=y. Sound right? What's the area?
 

1. How do you calculate the work required to empty a water-filled trough?

To calculate the work required to empty a water-filled trough, you need to know the weight of the water, the vertical distance from the water level to the top of the trough, and the acceleration due to gravity. The formula for work is W = mgh, where m is the mass of the water, g is the acceleration due to gravity, and h is the height of the trough.

2. What is the unit of measurement for work in this calculation?

The unit of measurement for work in this calculation is joules (J).

3. Can this calculation be used for any type of trough?

Yes, this calculation can be used for any type of trough as long as the three variables (weight of water, height of trough, and acceleration due to gravity) are known.

4. How does the weight of the water affect the amount of work required?

The weight of the water directly affects the amount of work required to empty the trough. The heavier the water, the more work will be required.

5. Is there a quicker way to calculate the work required to empty a water-filled trough?

Yes, if you know the volume of water in the trough and the density of water, you can use the formula W = mgVρ, where m is the mass of water, g is the acceleration due to gravity, V is the volume of water, and ρ is the density of water. This method may be quicker if you have access to this information.

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