Welcome to the first installment of wasteland survival with Dr. Albert (WSDA)! Throughout this series we will provide you with insights into surviving and thriving in this radioactive world we call home after the great nuclear war we’ve experienced. The occupants of Vault 76, myself included, have gone out into the hills of West Virginia. In my attempt to rebuild society and save humanity, I am creating the lessons on how to live in this new world of ours. This information is also available in holotape form through the link, and embedded below.

**Food**

Aside from water, the most important resource you will need to have available to you is food. We, therefore, begin our first installment of this series talking about getting food. When you first left your vault, you would likely have had to scavenge and find old non-perishable food in abandoned homes, stores or other areas. If you were unable to do this, you wouldn’t have made it to find these holotapes. Therefore, we will start by talking about hunting food once we have a weapon to do so with.

One of the most abundant animals in Appalachia are mole rats. These rodents are everywhere. Furthermore, the reproduce as rapidly as there prewar predecessors, rats. Considering that these mole rats are vastly larger than rats were, they provide a very reliable and sustainable source of food. I have found a nest that I visit regularly in order to harvest meat.

A few notes before you head in, these rats bite. Be careful and have a weapon ready not only to get your hunt, but also in order to defend yourself. You need to keep an eye open for more mole rats as you fight.

**One was getting away**

After clearing out the initial nest, there was one mole rat that got away. That is food that we could eat, so I want to chase after it. However, it also requires an extra expenditure of energy in order to chase down this mole rat. If I’m not going to get it, it’s not worth the energy. Even though I wouldn’t be able to do these calculations in the moment, if I prepare for this scenario before going on my hunt, I can know if it’s worth the energy.

In this case, I noticed a mole rat at the bottom right of the picture near the bottom of the billboard. Therefore, I want to see if I’ll catch it.

**Known information**

In this scenario, I know that I can run 6 mph and the mole rat can run 4 mph. I will run across the bridge toward the end of the billboard until I reach the billboard. The mole rat, however, will run diagonally on the ground. In order to simplify the problem, I will determine the rate of change based on the assumption that the mole rat was standing still. If we then add its speed to the result, we will get the total change.

We now let \(w\) be the width of the bridge, \(h\) be the height of the billboard and \(d\) the distance between me and the mole rat. This gives us the following picture.

We don’t know the distances \(w\), \(h\) or \(d\), so we will need some way to find this. In order to do this, I will approximate the height of the billboard by comparing it to my height. In the picture below, we see that I am slightly shorter than one of the sections of the board. Above, we see that there are 4 such sections. I am 5’7″, so we will assume each section is 6 ft. Therefore, the billboard is approximately 24 ft tall.

In order to find \(w\), we compare this to \(h\) by copying the length \(h\) onto the length \(w\). We see that \(w\) is approximately \(2.25h\), so we let \(w=54\)ft.

If we add in the information about the rates of change, we note that we know that \(\frac{dw}{dt}=-6\)mph. This is negative because \(w\) is decreasing. We are then trying to find \(\frac{dd}{dt}\), so we leave this as a question.

Now that we have the information provided, we need to find an equation relating the variables we know and that we don’t know. In particular, we need to relate \(d\) and \(w\). The triangle we created is a right triangle, so we can do this with the Pythagorean theorem. We then have that \(w^{2}+h^{2}=d^{2}\).

With this, we can now find \(d\) at the current point in time. We find that \(d=\sqrt{54^{2}+24^{2}}\approx 59\)ft. Furthermore, we have distances in feet and miles, so we will convert these in order to stay consistent. We get that \(6\)mph is \(6*1.47=8.8\)ft per second.

**Differentiating**

Now that we have all of our information compiled and converted into consistent units, we can find the rate of change between me and the mole rat. In order to do this, we need to implicitly differentiate the equation above. We then get \[\begin{align*} \frac{d}{dt}(w^{2}+h^{2})&=\frac{d}{dt}d^{2} \\ 2w\frac{dw}{dt}+2h\frac{dh}{dt}=2d\frac{dd}{dt}.\end{align*}\] Now that we’ve implicitly differentiated, we can insert the information we found above. We then get that \[\begin{align*} 2(54)(-8.8)+2(24)(0)&=2(59)\frac{dd}{dt} \\ \frac{dd}{dt}&=-8.05 \end{align*}.\]

Converting this back to miles per hour, we find that we are getting closer to the point the mole rat was at by 5.5 mph. Combining this with the speed of the mole rat, we note that we are then getting closer to the mole rat at a rate of 1.5mph.

**Conclusion**

We found that we are indeed able to make up ground on the mole rat. However, the speed at which I am catching up is not as high as the difference in speed of 2 mph. As we continue to move, despite the fact that both me and the mole rat will run at the same speed, we will have that rate of change between us will change. Therefore, we will have to spend more time looking at what happens in order to see if we do catch the mole rat. We will continue our work next time so, make sure to tune in for the next installment of WSDA to see what happens.

**Post Script**

I hope you found this post both entertaining and educational. In past posts, I have looked at how to incorporate video games as motivation for problems. However, I have not attempted to set my problems within the video game world itself before. Additionally, I have posted this same information in video form on YouTube in case you would rather watch a video than read. The goal is to provide examples of word problems, in this cases related rates, in a context that more people will enjoy. As such, having a written and video version seemed appropriate as it would reach different audiences. If you are a teacher, or a student learning calculus, I would greatly appreciate feedback on the attempt. Please let me know in the comments below, or by contacting me directly if you found this helpful. I plan on continuing this series, so any feedback will directly be used to improve future posts. Thank you.

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