Science:Math Exam Resources/Courses/MATH101/April 2010/Question 4
|Full-Solution 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
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
|Split the cone up into infinitely small cylinders and add up the volume of these cylinders using integration.|
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
|As the hint suggests, we split the cone into cylinders of height say Δx which are a height of from the ground (where the is an arbitrary sample point)|
In our diagram below, we have set it up so that the origin is at the apex (another way to solve this problem is to set up the 0 value at the bottom of the anthill). We now have, since the volume of a cylinder is πr2h where r is the radius of the base and h is the height of the cylinder, that
work done = weight distance
weight = density volume
work = density volume distance
where the distance is . Hence we can get the total volume by adding the contributions from each of the cylinders. In the limit as Δx goes to zero this becomes and integral and we get
Let us do a quick sanity check on the units to see if we have missed out anything.
The volume is in ft3, and the density is given in lb / ft3, 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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