A semi-sphere of sludge has a volume of 355cm³ and has a radius of 6.75cm at the top.  It takes 22.65seconds to consume.   What is the rate at which the height of sludge is changing at the moment when the sludge is 2cm from the bottom of the semi-sphere?

The two assumptions we make to solve this problem are:

1. The sludge is being consumed at a constant rate.

2. The bowl containing the sludge is a perfect semi-sphere.

A snapshot of this scenario is shown below:

Known Quantities at instant:

volume= 355cm³

t=22.65 sec

r=6.75cm

h=2cm

Our first step is to write an equation relating the quantities in the picture, so we will use the equation for the volume of a sphere:



Next we will use implicit differentiation with respect to time, in order to write a differential equation:

Unknown Quantities:

dV/dt

dr/dt

Since we are solving for dr/dt, we need to first find dV/dt

So, now we can revise our known quantities as such:

dV/dt=940.4cm³/min

r=2cm

Now, we can insert our known quantities into the equation in order to solve for dr/dt.

Finally, here is the graph of the function dr/dt:

This graph shows that when the sludge is at 2cm the rate of change the height of the sludge is 18.7cm/min, thus confirming our math was correct.

So, In words:

If prehistoric sludge is being consumed from a semi-sphere at a rate of 940.4cm³/min, then at the instant the sludge is 2cm from the bottom of the bowl, the rate at which the height of the sludge is changing is 18.7cm/min.

REFLECTIONS:

Tubby's Reflection:

At first I was skeptical about my related rates project, I simply thought we wouldn’t get it done. I was very happy to see it work out perfectly in the end. My roll in the group was data collector, organizer and equation solver. I cannot take all of the credit for the equation solving Freddy and John were both helpful as well. At first I felt like organizer of the entire project, but once we sat down as a group we all worked equally hard to create an (hopefully) excellent presentation and Wiki solution.

I learned a lot while working on this project; first I learned how to utilize working time with two very distractible teenage boys, at first we weren’t getting anywhere but when we sat down to do the whole thing together on Wednesday, we powered through the project and virtually finished it by the afternoon. This lesson will serve me well in the future because often in the past I have not been able to complete group projects and I imagine I will be pared with a few more boys by the end of my CRMS career. I also learned a lot about related rates problems. While we were learning how to solve them in class I was a little confused about which steps to use for different scenarios. The project helped with my understanding of the solving process because I was able to do the problem in reverse, which helped really analyze each step in stead of looking at it as one big problem (which makes me very confused and intimidated).

I had a lot of fun working on this group project; it really helped me understand the related rate concepts and was also a fun challenge that required a little creativity. I thought working with my group would be a pain, but we were all so excited about the problem in the end that we turned out to be a great, fully functioning group.

Winn Plot's Reflection:

     Overall I am absolutely thrilled with the results of this project. I think we create a really good concept and executed it perfection as well. I would have to say that each member of the group was a positive contributor and made sure that the math and the presentation of it was done correctly. My role to the group was consistently going through and searching for mistakes, whether it be in the mathematics or in the presentation. I tried to make sure that the math followed all of the proper steps that we practiced in the take home quiz, and held up to logical order of operations. Within the presentation portion I attempted to catch all of the spelling and grammar mistakes, as well as keeping our units consistent throughout. 

     What I gained from this project was a really applicable, tangible understanding of this concept of related rates, and I would say calculus as a whole. By applying it to a real world scenario, regardless of how rediculous it may have been, I was able to see it applied providing a greater sense of understanding. I really think I'm not only becoming more comfortable with calculus, but I'm also seeing further how it is the mathematics of change. I also appreciated the opportunity to work with geometric equation, such as the one we used, the volume of a sphere. This provided a further application for me and another area of tangibility, something that I feel really enhances my education and understanding. Lastly I learned a great deal about time management and handling problems, whether they be technical or attention based. We were able to come together as a group on Wednesday and really focus, work together, putting  together a project that I feel is a quality piece of work and something I can be proud of.

Babar's Reflection:

I felt that this project was a real good group effort.  Everybody contributed differently and equally.  My main contribution to this project was the work on the computer, in terms of  transferring the work we'd done onto the wiki.  Like I said before, we worked on almost all of this together, but because I had my laptop and because I'm familiar with all the computer-nerd related things, the majority of my contribution was computer related work.  I wasn't expecting everything to come together so nicely with this group, but in the end it did.  Joan and John helped out a lot, and when we all worked together and efficiently, we got a lot done. 

Through this project, I firstly learned more about related rates problems and how to go about solving one.  By creating, acting out, and then solving one, you really get a firm grasp on the subject.  I think the fact that we then have to present it and put it in the wiki, just forces us to be better experts of the subject.  I also learned how to efficiently and easily use the wiki, and I think just throughout the quarter the whole online thing has been getting clearer and easier to navigate.  I've worked in groups with Joan before, and many times, we have been very unproductive together.  I think the importance of this project really forced me to be much more focused and on-task, and so I learned that it doesn't matter how distractable me or my partner is, because if we just put our minds to it, and focus on staying focused, we can do really well.

I thought the end result was a great success.  We put a good amount of time and effort into this project, and we're all pretty happy with how this turned out.  Overall, this was a really good group project with good collaboration, as well as giving us a solid knowledge of the subject.