Is there a tank of the same size and same basic shape that would require less work to empty? Explain/justify your answer.
Is there a tank of the same size and same basic shape that would require less work to empty? Explain/justify your answer.
The work required to empty a tank depends on its shape and the height over which the fluid needs to be pumped. To determine whether there is a tank of the same size and same basic shape that would require less work to empty, we need to consider the characteristics of the tank's shape and size.
In the original given problem, we were given a specific tank with a cross-section described by the function y = f(x) or x = g(y), which determined the shape of the tank. The tank was 9 meters wide and 12 meters high. The work required to empty this tank was calculated based on this specific shape.
If we want to find a tank with the same basic shape but that would require less work to empty, we can consider the following possibilities:
** Wider Tank: If we keep the height of the tank the same but make it wider (increase the width), then the amount of fluid that needs to be lifted to the top edge would be less for the same height. This reduces the work required to empty the tank.
**Less Sloped Tank: If we modify the shape of the tank's cross-section (y = f(x) or x = g(y)) in such a way that it has a shallower slope (i.e., less variation in y over the same x range), then the work required to pump the fluid to the top edge would be reduced. Tanks with more uniform cross-sections will generally require less work.
**Shorter Height: Reducing the height of the tank while keeping the same width would also reduce the work required, as there would be less vertical distance to pump the fluid.
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