Science:Math Exam Resources/Courses/MATH101/April 2008/Question 02 (d)

MATH101 April 2008

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Question 02 (d)
Full-Solution Problem. Justify your answers and show all your work. Simplification of answers is not required. A cable that weighs 2 lb/ft is used to lift 800 lb of coal up a mine shaft 500 ft deep. Find the total work done (including the work done in lifting the cable itself).

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
Break this problem up into two parts. Compute how much work is done to lift up the coal. Then compute how much work is done to lift the cable up.

Hint 2
Remember that Work = Force x displacement.

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. A picture is included below. Let $x_{i}^{*}$ be a sample point. Then the work done on this ith part in foot-pounds is $W_{i}=2\Delta x\cdot x_{i}^{*}$ where the force is the $2\Delta x$ part and comes from the fact that since the cable weights 2 pounds per foot and the delta x is how much of the cable we are using in the ith part. The $x_{i}^{*}$ part is the distance. Hence, the work done is $W_{cable}=\lim _{n\to \infty }\sum _{i=1}^{n}2x_{i}^{*}\Delta x=\int _{0}^{500}2x\,dx=\left.x^{2}\right|_{0}^{500}=250000$ However, this does not include the work done to lift the coal itself. This undergoes $W_{coal}=800*500=400000$ and so the total work done is $W_{total}=400000+250000=650000$ where this is measured in foot pounds. This completes the proof.

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