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

FullSolution 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.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. A picture is included below. Let be a sample point. Then the work done on this ith part in footpounds is
where the force is the 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 part is the distance. Hence, the work done is
However, this does not include the work done to lift the coal itself. This undergoes
and so the total work done is
where this is measured in foot pounds. This completes the proof. 