The Giant Soup Can Problem

 

Write the given information: 

Look at the photo of the fallen water tank with the bike. We choose the dimension of the water tank (height or diameter) that would produce the minimum error. The dimension chosen should be distorted the least, and is parallel to a bike dimension (for example, height). This dimension and bike dimension should be close to each other on the camera line of sight. 

Suppose we choose the diameter. From photo we measure and obtain D/(h_b) which is 
(Diameter of the tank)/(Height of the bike) value using a ruler. Then, following computations estimate diameter, height and volume of water tank.

We can find, or construct and agree on, what are acceptable definitions of average house fire and required water amount (volume) to put out the fire. Let (V_ f) be required water volume to put out average house fire. Then,

Expanding the idea: Let us work on the same water tank with student birds in class. Given the photo of the fallen and little bit distorted tank with a bike, how do we estimate the dimensions and volume of the water tank?

Some students would want to organize the information we already have first.

 What we have:
1) The water tanks have the exact proportion of the Campbell’s soup can
2) The photo showing the fallen distorted tank and a bike

A student asks what is proportion? Another student answers, “It is diameter to length ratio”. When a key moment arrives, teacher steps in and reminds that we need clear statement written in completion not to make confusion. The variable of interest is volume of cylindrical water tank. If V is volume of water tank, v is the volume of soup can, then

It sounds like teacher forgets something. A student in the class has parents working in sheet metal business. This student points out that we are forgetting the thickness of the shell material used in construction of water tank and soup can. Someone recommends that we add assumption.

 Assumption 1: Thicknesses of the materials are negligible compared to the volume.

 What is next? Students shout out we need that proportion. Some students argue that we do not have a road map. We have not got a plan yet. Then one says we are planning now. Someone quietly listening now suggests that it is all good because we are in an adventure. We argue, point out weak point, but everything is peaceful. Finally a constructive idea is reached.

 We would copy, cut and match the water tank photo and the actual picture of the soup can. Another debate erupts on “what to match and why?” The tank in the photo looks distorted. A student asks if we can match the size of the letters because these letters seem to be the most intact in the photo. Another student agrees but adds assumption.

Assumption 2: The proportionality includes letters on the label. That is,

Students use school computer to do some image works. They use zoom in and out while keeping the proportionality. They compare, and finally they manage to get the letters on the tank and soup can nearly the same size. Therefore, the original soup can picture serves as original water tank alongside the damaged fallen water tank with the bike.

If the assumption (2) is true, this picture surprises the teacher. The water tank in the photo is distorted significantly. The answer quietly calculated by the teacher before seems to be not realistic now. This would become another key point that ‘looking from different points of view’ is beneficial. Teacher decides to discuss this after the project.

Students take measurements on the final picture and compute the ratio H/L and D/L. Now we need more information. We need the actual length L of the bike. Students arrive at a mid-point conclusion that we need the length of the bike to get water tank dimensions. Therefore, the class adds a new assumption.

Assumption 3: We have the length L of the bike.

Students then do the computations:

We do not even need the actual dimensions of the soup can. We also have a student whose parents work in some engineering design jobs. This student remembers that our assumptions may be incorrect and we need to inspect. When the student points that out, the student is not very sure what exactly that is. Teacher catches another key moment to explain what is in the student mind.

Sensitivity analysis:
If a value in the computation is actually different from what we assumed, then the final result can change. What is the effect of a certain amount of variation in data on the final result?

Before the teacher bird opens her mouth to confess her prior solution and its weak points, student birds demand for explanation why we need so-called sensitivity. Instead of answering directly, teacher asks if they want their firefighting water tank to be smaller than the necessary size? This sparks another round of discussion for importance of responsible reporting of computational results in real life. This class project produces some important points:

• Assumptions must be made after reasonable amount of unbiased inspection
• Statements (including reporting) should be produced in completion to avoid misunderstanding
• When seeking a solution (in this case minimum error estimate), past experience and different points of views or methods should be applied
• Need of experiments, exploration, and learning of skills and reasoning required by the problem at hand (in this problem, knowing that two images need to be at the same state and actually making it happen)
•  Possible weak points (for example due to assumptions) of each method should be discussed
  
• Sensitivity analysis on all assumed values should be done
• Responsible reporting

Additionally, if we do not have a means to manipulate the pictures (advantage of technology), we would still need the actual dimensions (letter size also) of the soup can. Finally everyone agrees the importance of sharing of diverse views, methods and knowledge; and need to continue learning to gain greater maturity (experience). It is almost 7 pm. It is decided that all birds need a rest. 

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