Solving Water Tank Problems with Maple/Mathematica

[Hollysmoke]

1. Consider a rectangular water tank with a length of 8m, a width of 4m , and a height of 5m. Water exits the tank by pouring over the sides.
a) If the tank is initially full of water, find the work required to empty the tank.
b) If the tank is initially half-full of water, find the work required to empty the tank.
c) If the tank is initially full of water, find the work required to pump out half of the water
d) If the tank is initially full of water, find the work F(y) required to pump out the water until the depth of water in the tank is y. Plot F(y)
e) If your goal is to pump out all of the water, at which water level y0 will half of the work be done?




2. How can I get this onto maple?



3. Done it by hand, just have no idea how to solve it on the computer =(

 

[cristo]

How do you have no idea how to do it on the computer? If you've done it by hand, then you know the equations you need to solve. Show your work!

 

[tacman]

Using Maple or Mathematica to solve water tank problems can greatly simplify the process and provide accurate results. These software programs have built-in functions and tools specifically designed for solving mathematical problems, making them ideal for tackling complex water tank problems.

To solve the given problem, we can use the built-in functions in Maple or Mathematica to calculate the volume of the tank and the amount of water in the tank at different stages. We can then use these values to determine the work required to empty the tank or pump out a certain amount of water.

a) To find the work required to empty the tank when it is initially full, we can use the formula W = mgh, where m is the mass of the water, g is the acceleration due to gravity, and h is the height of the water. Using the built-in functions in Maple or Mathematica, we can easily calculate the mass of the water and then use the formula to find the work required.

b) Similarly, to find the work required to empty the tank when it is initially half-full, we can use the same formula and calculate the mass of the water at this stage.

c) To find the work required to pump out half of the water, we can first use the built-in functions to calculate the volume of the tank and then find half of this volume. We can then calculate the mass of this amount of water and use the formula W = mgh to find the work required.

d) To find the work required to pump out the water until the depth of water in the tank is y, we can use a similar approach. First, we can use the built-in functions to calculate the volume of water at a depth of y. Then, we can calculate the mass of this amount of water and use the formula W = mgh to find the work required. We can repeat this process for different values of y and plot the results to see how the work required changes as the water level decreases.

e) To find the water level at which half of the work is done, we can use the results from part d) and plot the work required as a function of water level. We can then use the built-in functions to find the point at which half of the work is done, which will give us the water level y0.

To solve these problems on Maple or Mathematica, we can use the "solve" or "fsolve" functions, which allow us to input equations and variables and find the numerical