Exercises / Discussion questions
(1) Scene 6 (initially) shows a tank of height filled with a liquid of weight density . The radius of the tank at height is , for (where and are both given in ). We're interested in first approximating, then computing exactly (at least relative to a simplified physical model) the work required to pump all of the liquid from the tank to a height of above ground level. View scene 6 and click the Animate pumping button to animate the pumping. Note that in the animation the liquid simply disappears after it reaches the required height. Would it be relevant to the work required if after reaching the height of the liquid was routed through a pipe parallel to the ground and then allowed to spill? (Assume the pipe offers negligible resistance to the fluid flow.)
(2) What if the pipe after a 9 ft horizontal segment turned downward and extended down into a reservoir at ground level? Explain your answer.
(3) What if the fluid was pumped into the bottom of a tank of the same size, shape, and orientation as the original tank, just located directly above the original tank? Explain your answer.
(4) Now consider a tank which is the "upside down" version of the original tank, except that the new top is now open and the new bottom is closed. (Click the Flip tank button if you wish to visual this.) Would it require more work, less work, or the same amount of work to empty the "flipped" tank as compared to the original tank? Explain.
(5) Next approximate the work required to pump the fluid to the desired height as follows. Divide the fluid in the tank into 4 layers of equal height. Approximate each layer as a right circular cylindrical disk whose radius is given by the radius of the tank at the bottom of the layer. Consider these disks to have the same weight density as the fluid. Approximate the work required for the original pumping problem by computing the work required to lift the 4 disks to a height of . Show the details of your computation. You can visualize this approximation by flipping back to the original tank (if necessary), then clicking Discretize followed by Animate lifting.
(6) What are the mathematical sources of error in the approximation above? (Ignore errors corresponding to our simplified physical model which neglects all effects arising from frictional forces.)
(7) Is your approximation from exercise (5) clearly an overestimate? Is it clearly an underestimate? Explain.
(8) Answer question (7) for the "flipped" version of the tank assuming the work approximation is computed analogously.
(9) Use MATLAB to compute an approximation following the method of exercise (5), but dividing into 100 layers instead of just 4. Copy and paste your MATLAB code as well as the result here.
(10) Set up an expression using sigma notation for the approximation that follows the method of exercise (5) and uses layers.
(11) Do the mathematical errors go to zero as goes to in your sigma expression? Justify your answer.
(12) Write out the definite integral represented by the limit as goes to in your sigma expression.
(13) Compute the value of the definite integral from the previous exercise.
(14) What's the relative error in the MATLAB approximation from exercise (9)?
