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Question 04 

FullSolution Problem. Justify your answer and show all your work. Simplification of the answer is not required. A colony of ants builds an anthill that is in the shape of a cone whose base, at ground level, is a circle of diameter 1 ft and whose height is also 1 ft. How much total work, in ftlbs, is done by the ants in building the anthill? For the density of sand, use the value 
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Hint 

Split the cone up into cylinders with infinitely small height and add up the volume of these cylinders using integration. 
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Solution 

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Consider the diagram for building the anthill . We can think of building the hill as stacking several small cylinders of height Δx or dx (thinking forward to integration) from the ground (x=1) to the top (x=0). Consider the anthill at some intermediate distance from the top x (at a distance 1x from the ground). This point is represented in the diagram as . The volume of a new cylinder to add onto the anthill will be
Please rate my easiness! It's fun and helps everyone guide their studies. and so Notice that indeed when we're at the top (x=0) then the radius is zero and when we're at the bottom (x=1) that the radius is 1/2. We therefore have that the new amount of sand volume we are adding is In the problem we are given a density 150lb / ft^{3} which is actually a force density because lbs is a measure of force. This is telling us the force required to lift sand of a given volume which is precisely what we are trying to do with our cylinder of sand and so the force required to lift it is force = force density volume The work required to lift this cylinder to its position in the hill at x will be this force dF multiplied with the height we have to lift it from the ground 1x. Therefore the amount of work to lift this cylinder is This is the work to build one small cylinder. The whole hill is built by adding cylinders for all values of x and so to get the total work we integrate Let us do a quick sanity check on the units to see if we have missed out anything. The volume is in ft^{3}, and the density is given in lb / ft^{3}, so the weight is in lb, finally work is force (weight) times distance (which is in ft), so we get our work to be in , which is right. In conclusion, the work done by the ants to build the anthill is . 
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