How to answer a modelling question
Modelling questions are always due to cause a bit of shock during your exam paper. Symptoms such as scratching heads and chewing pens are strong indicators that a student is deep into a modelling question. These questions require your mathematical knowledge is strong across topics and you are able to decipher what the question is asking. The former is achievable through dedicated bookwork; however, issues often arise for the second as students are unsure what to apply when given an abstract statement, rather than a simple “Rearrange to solve for 𝒙”.
Give me an example
A diver searching a shipwreck on the ocean floor is beginning her return back to the surface. Her motion is governed by the parametric equations:
where 𝒙 is the horizontal distance travelled east and 𝒚 is the vertical distance travelled, both measured in metres.
There is a shark located at 𝟤𝟪𝟪𝟢𝒎 east along the horizontal direction of the diver’s journey. If the diver hits this location outside of the range
-𝟣𝟫𝟤< 𝒚 <-𝟣𝟤𝟢, they will safely make it past undetected. Does the diver pass through safely?(𝟥 marks)
Okay, so here we are presented with a situation that doesn’t read quite as straightforward as “Here’s 𝒂, here’s 𝒃. Go find 𝒄”. Instead we’re given a few sentences about a diver, a shipwreck and a shark and that if some condition is met, our diver will avoid running into the shark. This is not particularly mathsy.
A starting point everyone should be comfortable with is picking out the maths that is blatantly given to us: we’ve got parametric equations given to us in the first sentence.
Nice. Then there is a bit of maths / English that needs translating in order to do anything else here. The next line gives us the translation we need, stating that 𝒙 is our horizontal displacement for distance travelled east, and 𝒚 is the vertical displacement. Now we have some context of what 𝒙 and 𝒚 represent.
In the final part of the question we are told the shark is 𝟤𝟪𝟪𝟢𝒎 east of the diver’s location, and that if she is within the range -𝟣𝟫𝟤< 𝒚 <-𝟣𝟤𝟢, then she will miss the shark. In other words, if the horizontal displacement is 𝟤𝟪𝟪𝟢𝒎, the diver has reached the 𝒙 coordinate of the shark. When the diver has this 𝒙 position, and her 𝒚 position is in that range -𝟣𝟫𝟤 to -𝟣𝟤𝟢, then she is safe: otherwise she is in the danger zone. At this point we can confidently label the root problem of this question to be coordinate geometry.
“Given that the diver has this 𝒙 coordinate, what is her 𝒚 coordinate?” is one way we could write it.
This is a great opportunity to add a sketch to your working. Visualising this helps verify working and give you some confidence in your method.
If you think you have a grasp on what to do next then give it a go if you can, or take a peek below to see what’s going on.
With some context now provided it should be clear we don’t need to look to forming a Cartesian equation using these parametric equations, simply notice both parametric equations share the time, 𝒕.
Substitute 𝒙 = 𝟤𝟪𝟪𝟢 into the parametric equation for 𝒙. Now we can rearrange for 𝒕.
This is the time at which the diver has reached the 𝒙 coordinate of the shark. Using this time in the parametric equation for 𝒚, we get the height at which the diver is, and thus see if she will be spotted by the shark.
Comparing this to -𝟣𝟫𝟤< 𝒚 <-𝟣𝟤𝟢 , we can see that she is not within this range, and as such the diver avoids the shark. Success!
The take away
Now, obviously this is just a taster example, and modelling questions can get far more complex, but it is a good illustration of how to translate instructions and additional information within a question that disguises itself through a situation. Above we were able to spot this was a coordinate geometry question and use the parametric equations accordingly. We also made a sketch to confirm that the maths we were doing actually made sense.
The issue with modelling problems is that they are so variable. The context can be complex and unique, meaning that each solution is hand crafted. This means that resources in textbooks can be limited and our go-to response of trying examples in the textbook don’t cover the nuances of how to actually answer these questions.
To truly ace modelling problems, the key lies in your ability as a mathematical detective and understand the information provided to you. Once you have this in a way that is easy for you to understand you can pick the correct maths tools for the job to solve.
At AITutor we are working hard to provide a one-stop shop for your study needs and are continually rolling out new questions and solutions like the example used in this blog to prepare you in the best way for your exams. If you want to find out more, check out our site!
