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

MATH101 April 2009

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Question 07 (b)
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/m3). (b) Express the work required to pump the water out of the tank, if it is pumped out over the top edge, as an integral (Do not evaluate the integral.)

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
Try drawing a picture first (in the solution, we will set the origin to be the top most point).

Hint 2
Use similar triangles to find the width y of a small rectangular prism at height $x_{i}^{*}$.

Hint 3
The following formulas will probably be useful Volume = length x width x height Mass = volume x density Force = acceleration x mass = gravity x mass = 9.8 x mass 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 at the bottom of the page. Let $x_{i}^{*}$ be a sample point. First, we use similar triangles as in the diagram to see that $\displaystyle {\frac {1}{2}}={\frac {y}{2-x_{i}^{*}}}$ Isolating yields that $\displaystyle y=1-{\frac {x_{i}^{*}}{2}}$ Now we use hint 3 and write down the necessary formulas. For volume of the ith piece, we have $V_{i}=y\Delta x\cdot 10=10(1-{\frac {x_{i}^{*}}{2}})\Delta x=(10-5x_{i}^{*})\Delta x$ For the mass of the ith piece, where $density=\rho =1000$, we have $m_{i}=V\cdot \rho =(10-5x_{i}^{*})\Delta x(1000)=1000(10-5x_{i}^{*})\Delta x$ For the force, where $g=9.8$, we have $F_{i}=m_{i}a=9.8(1000(10-5x_{i}^{*})\Delta x)=9800(10-5x_{i}^{*})\Delta x$ For the work on the ith piece, we have $W_{i}=F_{i}\cdot d=9800(10-5x_{i}^{*})\Delta x(x_{i}^{*})=9800x_{i}^{*}(10-5x_{i}^{*})\Delta x$ and so the total work done is $W=\lim _{n\to \infty }\sum _{i=1}^{n}W_{i}=\int _{0}^{2}9800x(10-5x)\,dx$ completing the question.

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