Science:Math Exam Resources/Courses/MATH104/December 2011/Question 04
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Question 04 

A conical tank of height of 5 metres and top radius 4 metres is filled with water and then drains into a cylindrical container of height 5 metres and radius 4 metres. If the water level in the conical tank drops at a constant rate of 0.5 metres per minute, at what rate does the water level in the cylindrical tank rise when the water level in the conical tank is 3 metres? The volume of a cone of radius r and height h is . The volume of a cylinder of radius R and height H is . 
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 

Write down everything you know about the problem. Try drawing a picture. 
Hint 2 

In order to solve this problem, you need to differentiate both volume formulas with respect to t. However, before you do so, you should rewrite the formula for the volume of a cone in terms of h, with no r. 
Hint 3 

Using similar triangles, we can determine that the relationship between the radius and height of water in the first tank is given by the ratio

Hint 4 

What is the relationship between the volume of water, and its rate of change, in the first and second tank? 
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. We begin by assembling all the information we know about the problem:
We note that for this problem, we are mostly concerned with change in height and change in volume, not change in radius. Therefore, we will rewrite our two volume formulas in terms of h as follows: First, we use fact #3 and solve for r in terms of h, or . We can then rewrite the formula given in #1 as a function of .
For the volume formula given in #2, the radius doesn't change (because the tank has straight sides) and equals 4. So we have a volume formula as a function of .
We now have two volume formulas in terms of h. Because the question is asking about change over time, we will implicitly differentiate with respect to t. This gives: for the first tank and for the second tank. If we consider the first formula, we realize from #5 that we know and furthermore, the statement of the question tells us that . We plug these into the first formula to get Now we remember from #4 that . So . Plugging that into our second volume formula we get Finally, we know from #6 that we're trying to find . Thus we simply solve the previous equation for to get: Which is our answer. 