Science:Math Exam Resources/Courses/MATH101/April 2005/Question 05 (a)
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Question 05 (a) 

A water tank with depth 3 feet is in the shape of the trough depicted below. The bottom edge of the tank is 1 foot above the ground, and the ends of the tank are equilateral triangles of side length feet; the top of the tank is a rectangle of length 12 feet and width feet. (a) The tank is filled with water pumped up from the ground level. How much work is done in filling the tank? Express your answer in footpounds, and use the value 62 lb/ft for the density of water. (Note that in the Imperial System, pounds are a unit for force.) (Do not simplify your answer.) 
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 

Imagine looking at the trough face on to the equilateral triangle. Choose a coordinate system so that 0 is on the top. Choose a sample point and consider the volume that the small rectangular prism creates. 
Hint 2 

The following formulas will probably be useful Volume of a rectangular prism = length x width x height Imperial Force = Imperial density x volume = gravity x mass = 9.8 x mass Work = Imperial 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 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. Consider the following diagram: In the picture, we take the perpendicular bisector of the top base and draw the line. This splits our equilateral triangle into two equal parts. The base is and the height is 3 and the other side length is . Choose a sample point say . Let's call the length of the small triangle point downward l. By similar triangles, we have
and hence
The volume of the thin prism at this height is
Next, the force acting on this prism is
Lastly, the work done on this piece, noting that the displacement is 1 added to the distance to the bottom of the triangle (which is )
Summing over all pieces, we have
and this last value is an integral given by
