After-Action Report: Rational Functions

First, let’s discuss the “mini-project” about the shadow. You were asked to graph the relationship between the height of a person (or, more generally, an object) and the length of her shadow. How can we approach this? Let’s look at the picture. What do we see? Anything that can help us do some math? How about a triangle:

Now, since we’re being asked to graph a relationship, we probably need to define a function; which means we need to define an input and an output. We’re being told what relationship to graph, namely, the relationship between a person’s height and the length of her shadow. Which one of these is the input and which is the output? Let’s think about that. Does a person’s height depend on the length of their shadow? Probably not. A person’s height depends on lots of things, like her genetic makeup, age, health, but not the length of her shadow. However, we can be pretty sure that the length of a person’s shadow depends on her height. Shorter people, in general, have shorter shadows. The means that the shadow length is dependent, which means it’s the output. The person’s height is independent, which means it’s the input. Typically we use x for the input and y for the output, so let’s label those in the picture.

Now let’s label the other parts of the picture. It seems like the distance from the wall and the height of the light are probably important, since they’re the only two values that aren’t already labeled. The hypotenuse might be important, but we know that we can use the Pythagorean theorem to define the length of the hypotenuse in terms of the legs if we need to.

Are there any geometric relationships we know about that can help us define an algebraic relationship? How about similar triangles? There are two similar triangles in the picture. Remember from Geometry that corresponding sides of similar triangles have lengths in the same ratio. So \frac{y}{y+d}=\frac{x}{h}. Solving for y gives us y=\frac{dx}{-x+h}, so our graph should be a rational function with a vertical asymptote at h, a horizontal asymptote at -d, and an intercept at the origin. It looks like this:

For the push pop project, you were asked to redesign ice cream push pops under three considerations; minimizing materials cost, minimizing thaw rate, and making the package more appealing to the consumer. Minimizing materials cost and minimizing thaw rate are essentially the same thing, since they both depend on surface area. We could use different materials that are either cheaper or better insulators or both, and a few groups came up with creative solutions involving different materials, but no matter what material we use, we will minimize thaw rate and materials cost by minimizing surface area. Making the package more appealing to the consumer can depend on a lot of things – unique shapes, ease of use… there are lots of ways to approach this question, but here I’m going to focus on minimizing the surface area. Push pops, for the most part, are cylinders. A few groups suggested other shapes, but I’m going to focus on cylinders. The same basic ideas would apply to other shapes. So, we need to minimize the surface area of a cylinder. Let’s start with the equation for surface area, A=2\pi r^2+2\pi r h. Unfortunately, this equation has two independent variables, r and h. In order to get a function that we can minimize, we need to get A in terms of r or h, which means we need h in terms of r or the other way around. Are there any other equations that relate r and h? The only one I can think of is the equation for volume, V=\pi r^2 h. You were told that you could treat V as a constant, so that means we can use this equation to get h in terms of r. Solving for h gives us h=\frac{V}{\pi r^2}. Now we can substitute this for h in the surface area equation. Doing so and simplifying gives us A=\frac{2\pi r^3+2V}{r}. Now we have A as a function on r, allowing us to minimize A. We should now draw a graph with r as the horizontal axis and A and the vertical axis but this is a pretty tricky rational function to graph. We know that there’s a vertical asymptote at r=0 and no A-intercept. We could find that the r-intercept is -\sqrt[3]{\frac{V}{\pi}}, but that’s not that interesting because it’s obviously negative and outside our domain. We know we don’t have a horizontal asymptote since the degree of the numerator is greater than the degree of the denominator. Some of you may have remembered that in this case we need to use polynomial long division to find the equation defining a slant or other type of asymptote. If you did this, you would find that the quotient is 2\pi r^2, meaning that our asymptote is in the shape of a parabola. Some of you just picked a volume and used a computer graphing device or picked test points, both of which were fine solutions. If we picked a volume of 100ml (about 3 1/2 fl oz), our graph would look like this:

The minimum is somewhere between r=2 and r=3. Wolfram Alpha tells us that it’s at r=5^\frac{2}{3} \sqrt[3]{\frac{2}{\pi}}, or about 2.5. Plugging this into our equation for h in terms of r gives us approximately 5.1. So a radius of 2.5cm and a height of 5.1cm would minimize the surface area for a push pop of 100ml. Would a push pop with these dimensions be appealing to the consumer? I’ll leave that for you to puzzle out.

What have we learned about rational functions? What kinds of phenomena do they model? How can they be graphed? Leave your thoughts in the comments.


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