A container filled with water with an outlet at the bottom can be used as a timer (similar to an hourglass). Use the principle of conservation of mass to make a mathematical model of the height of the water in a conical container as a function of time, h(t). The diameter of the outlet is d and the slope of the cone is specified by the angle theta, as shown on the schematic below. Assume that the average outlet velocity is proportional to the height of the water according to the equation V = squareroot 2gh. Find an equation for the required height of the container, H, if it is to empty in T seconds. If it is to take 10 minutes for the container to empty, plot the required height H as well as the diameter of the top of the cone, D, as functions of the angle theta for 15 degree lessthanorequalto theta lessthanorequalto 75 degree. Assume d = 1 cm. Use Excel to make the plots and put H and D on the same plot. Properly label the plot. Turn in all of your work and your plot, as well as a brief discussion about the results. Discuss what you believe would be the best angle theta and why.
A closed rectangular container with a square base is to have a volume of 20002000 cubic centimeters. It costs twice as much per square centimeter for the top and bottom as it does for the sides. Find the dimensions of the container of least cost.
So the formula for the volume is: 2000=x2y2000=x2y and formula for cost (area) is 4xy+4x24xy+4x2. After some substitutions and taking the derivative of the cost formula, I found that there is one critical number for x=10x=10, and substituting back to volume formula, I found that y=20y=20.
But now, i need to show that 1010 is the absolute minimum, I\'m confused about how do I need to show that? Just plug in back to the original cost formula? It will just give me another number, which does not help at all.
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