Science:Math Exam Resources/Courses/MATH101/April 2011/Question 03 (a)

MATH101 April 2011

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Question 03 (a)
A tank in the shape of a vertical circular cylinder 2 m high and with radius 0.5 m is filled with a fluid that has density $1000kg/m^{3}$. Write a definite integral that gives the work, in Joules, required to pump all the fluid to the level of the top of the tank. For acceleration due to gravity, use $g=9.8m/s^{2}$. 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
Draw a diagram of the tank and see if you can interpret the question with the help of that diagram.

Hint 2
Recall that the work necessary to displace an object of mass m over a distance d is W = mgd.

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. To solve this problem, we consider moving up the liquid in the cylinder slice by slice. Consider the slice of the cylinder of height dx as in the picture. Its volume is given by A dx where a is the area of the disc. This is $\pi (0.5)^{2}\,dx$ Its mass is thus $1000\pi (0.5)^{2}\,dx=250\pi \,dx$ Since the density of the liquid is 1000 kg/m3. The work to lift the slice to the top is force multiplied by distance which is $9.8\cdot 250\pi \,dx\cdot x=9.8\cdot 250\pi x\,dx$ where x is the distance from the top of the cylinder to the slice. Hence the total work is given by integrating this from the top to the bottom of the cylinder, that is from 0 to 2 and we obtain the integral $W=\int _{0}^{2}9.8\cdot 250\pi x\,dx$

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