Solving Water Tank Homework: Find Work & Remaining Depth

In summary: The units for acceleration due to gravity can be either m/s^2 or ft/s^2, depending on which units you are using for the rest of the problem. In this case, since the problem uses feet, you would use ft/s^2.
  • #1
Gauss177
38
0

Homework Statement


A tank full of water has the shape of a parabloid of revolution with shape obtained by rotating a parabola about a vertical axis.
a) If its height is 4 ft and the radius at the top is 4 ft, find the work required to pump the water out of the tank.
b) After 4000 ft-lb of work has been done, what is the depth of the water remaining in the tank?

Homework Equations


m = density*volume

The Attempt at a Solution


I don't know how to do part (b). This is what I have for (a):
I labeled the radius of cross section as Ri (ith subinterval)
Ri/(4-Xi) = 4/4
Ri = 4-Xi
Volume of ith layer of water = pi(4-Xi)^2 dx
Mass of ith layer of water = 62.5pi(4-Xi)^2 dx
Force to raise ith layer = (9.8 m/s^2)(62.5pi(4-Xi)^2 dx
W to raise ith layer = 612.5pi*x*(4-x)^2 dx
Total work = Integral of 612.5pi*x*(4-x)^2 dx on [0, 4]

The answer is not right, so can anybody tell me what I did wrong and how to fix it? Also, how would you do part (b)?

Thanks :smile:
 
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  • #2
You seem to have got the wrong shape for the tank. The equation of any parabola is y = ax^2 (with y vertical and x horizontal) and the problem says y = 4 when x = 4 so you can find the value of a.

Your "Ri = 4-Xi" seems wrong - that would be a cone, not a parabola.

In your "W to raise ith layer" you are not using consistent units - you used g in m/s^2.

For part (b), just find the work to pump out the water to depth D (a similar integral to the first part).
 
Last edited:
  • #3
I see. What do you do with the 'a' if I plug in 4 for both x and y? I get a=1/4, but not sure where to go with that.

I thought of it as a cone, so that was wrong. How would you find the radius then for a parabola?
 
  • #4
You know y = 1/4 x^2, so rearranging that, x = 2 sqrt(y)

Or in words: at height y above the base of the tank, the radius is 2 sqrt(y).
 
  • #5
thanks. But instead of using m/s^2 for acceleration of gravity, what should the units be? The problem uses feet, so is it ft/s^2? ft/h^2? I'm not sure.
 

What is the purpose of solving water tank homework?

The purpose of solving water tank homework is to determine the amount of work required to empty or fill a water tank, as well as the remaining depth of the water in the tank at a given time.

What equations are necessary for solving water tank problems?

The equations necessary for solving water tank problems include the volume of a cylinder (V = πr²h), the formula for work (W = Fd), and the principle of conservation of energy (PE + KE = constant).

What information is needed to solve a water tank homework problem?

To solve a water tank homework problem, you will need to know the dimensions of the tank (radius and height), the initial depth of the water, the flow rate of the inlet or outlet, and the time elapsed.

What are the steps for solving a water tank homework problem?

The steps for solving a water tank homework problem include setting up the given information, calculating the volume of water in the tank at different time intervals, finding the work done to fill or empty the tank, and using the principle of conservation of energy to determine the remaining depth of water in the tank at a given time.

Why is solving water tank homework important?

Solving water tank homework is important because it allows us to understand the mechanics of filling and emptying a tank, and also helps us to calculate the work involved in these processes. This knowledge is useful in real-world applications such as designing efficient water storage systems or understanding the energy requirements for pumping water.

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