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 cylinders with infinitely small height 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.
| or (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 1-x from the ground). This point is represented in the diagram as . The volume of a new cylinder to add onto the anthill will be
Consider the diagram for building the anthill |
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 which is actually a force density because lbs is a measure of force (note that this means, unlike for a mass density, we do not need to multiply by acceleration). 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 1-x. 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 , and the density is given in , so the weight is in , finally work is force (weight) times distance (which is in ), so we get our work to be in , which is right.
In conclusion, the work done by the ants to build the anthill is .