Science:Math Exam Resources/Courses/MATH103/April 2010/Question 06 (b)

MATH103 April 2010

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Question 06 (b)
The height of fluid, $h(t)$ in a cylindrical container is controlled by a pump so that it satisfies the differential equation ${\frac {dh}{dt}}=-kh^{1/3},\quad h(0)=h_{0}$ where $k>0$, $h_{0}>0$ are constants, and $h_{0}$ is the initial height of the fluid. (b) At what time will the container be empty?

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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
The container is empty when the height of the fluid in the container is zero.

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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 find the time tempty when the height h(tempty) is zero we calculate $0=h(t_{\text{empty}})=\left(h_{0}^{2/3}-{\frac {2}{3}}kt_{\text{empty}}\right)^{3/2}.$ Solving for tempty yields $t_{\text{empty}}={\frac {3}{2k}}h_{0}^{2/3}.$ Note: The differential equation is no longer valid after this time, because a negative height of fluid in a tank does not make sense.

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