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Question 08 (a) 

Consider two tanks of water, one above the other, as pictured below. The top tank, which is initially full, is a cylinder of radius and height . The bottom tank, which is initially empty, is a cone of radius and identical volume as the top tank. Suppose water drains at a constant rate from the top tank to the bottom tank. (a) Find the rate at which the depth of water in the bottom tank is increasing when it is equal to . Your answer should be in terms of and . 
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 

Since water drains at a constant rate , it means that the rate of change of the volume of water over time is the constant. 
Checking a solution serves two purposes: helping you if, after having used the hint, 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. Denote the depth of the water in the bottom tank by and the radius of the cone filled with water with the depth of by . Then, using the ratio between the height and the radius as in the picture on the right, we get .
By the Hint, the rate of change of over time is constant. i.e., . Taking a derivative on both side of the equation with respect to , we have In the second equality, the derivative only hits because and are fixed constants. (i.e., independent of time.) Therefore, at the depth , the rate is .
