Science:Math Exam Resources/Courses/MATH101/April 2009/Question 07 (a)
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Question 07 (a) 

An open metal tank has two ends which are isosceles triangles with vertex at the bottom, two sides which are rectangular, and an open top. The tank is 1 metre wide, 2 metres deep, 10 metres long and full of water (density = 1000 kg/m^{3}).

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 

Recall that hydrostatic pressure is given by, where is the density of water, g, the gravitational constant, and h the height from the top. It is important that height is measured from the top of the surface because as depth increases, there is more water above you, increasing to the pressure. 
Hint 2 

Recall that they hydrostatic force on an object of area A at height h is where is the density of the surrounding liquid. 
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. Hydrostatic Force was not covered in the 2012 offering of this course. Time might better be spent solving other problems given how soon the exam is coming.
We could then get the total force by integrating dF but first we need an expression for the length in terms of the height, y. The length of any rectangle will start at the left edge of the triangle and end at the right edge. We can use the coordinates that we listed to find equations for these lines. The right edge is given by which written in terms of y is, where we have added a subscript R to indicate it's the right edge. Notice we compute the line by finding the slope using the right vertex and bottom vertex and then notice that the line goes through our origin to its yintercept is zero. We can similarly get that the left edge is This can also be argued with symmetry of the right edge. We then have that Therefore we can now get F via integration where we see that y starts at zero and ends at y=2. Therefore, We are told in the problem that the density is 1000kg/m^3 and we take the gravitational constant as 9.8N/kg. Therefore, This is the hydrostatic force on each of the triangular sides. 