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course Mth 173
9/30 18:30
QuestionIf a sand pile 4 meters high has a mass of 146000 kg, then what would we expect to be the mass of a geometrically similar sand pile 14 meters high? Using the differential estimate the mass of sand required to increase the height of the pile from 4 meters to 4.03 meters.
Answer
The volume of a sand pile is directly proportional to the cube of the height of the cone and since density = mass / volume, thus we also know that mass is directly proportional to the volume and so the mass of the sand pile will be directly proportional to the cube of the height of the cone.
Thus m = k * (h^3), where h is the height (altitude) of the cone and m is the mass and k is the proportionality constant.
We know that for a 4 meter high cone the mass is 146000 kg thus substituting the values in the function we obtain the value of the proportionality constant.
146000 kg = k * (4 ^ 3 cubic meters) = k * (64 cubic meters)
Thus k = 146000 kg / 64 cubic meters = 2281.25 kg / cubic meters
We know the proportionality constant of the function and to find the mass of the sand pile of height 14 meters can be found by substituting the value of a and h.
Thus mass of 14 meters high sand pile
m = 2281.25 kg / cubic meter * (14 ^ 3 cubic meters) = 2281.25 kg / cubic meters * 2744 cubic meters = 6,259,750 kg
The function given is m = k * (h ^ 3)
Differentiating the mass function for height we get the rate of change of mass with respect to height
m’ = 3k * (h^2)
m’(4) will give the rate of change of mass when height of the sand pile is 4 meters. Now since 4.03 meters is very close to 4 thus we may consider the rate of mass change to not vary much in that interval of 4 and 4.03. thus the average rate of change of mass in the 4 and 4.03 meters interval may be considered to be m’(4)
m’ (4) = 3 * 2281.25 kg / cubic meters * (4 * 4) square meters = 109,500 kg / meters
We also know that average rate of mass change = change in mass in that interval / change in height in that interval
Change in height in that interval = 4.03 meters - 4 meters = 0.03 meters
Thus change in mass in that interval = average rate of mass change * change in height in that interval
= 109500 kg / meters * 0.03 meters = 3285 kg
Thus this is the estimate of the additional mass required to change the height of the sand pile to increase from 4 to 4.03. This is the estimate because the actual rate may vary in that interval but the variation is very small and thus can the value can be estimated. _________
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Excellent work, excellent explanation.
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