**Modeling the Price of Different Sized TVs**

It really started with a joke.

“Oh, you need a new TV? You should get a 70 inch. Wait, what about the 80 inch TV? Well, those quite a bit more expensive than the 70 inch TV”

I really hadn’t looked carefully at TV prices so I wasn’t aware that the price increased dramatically with size. I’m aware now.

You can probably guess what comes next. Yes, I need to go to Amazon and look up the prices of different sized TVs. Now, there is a problem comparing TVs of different sizes. You could have a 42 inch TV with more features than a 60 inch and that could have an impact on the price. What I need are TVs that are the same model except for size. Fortunately, I was able to find some models just like that from Sharp, Samsung and Vizio. Along with this, I have the lowest prices for the different size ranges as advertised by the Walmart website.

The different brands seems to be similar enough that we can look at this all as one set of data. So, what happens when you double the size of a TV? Does the price also double? No. A 40 inch TV is much less than half the cost of an 80 inch TV. Well maybe the size is proportional to the area of the TV.

First, let’s look at the listed “size” vs. the actual area. The standard measurement for a TV is to give the diagonal distance from one corner to the opposite. We also know that a standard HDTV has a 16:9 aspect ratio. Maybe this diagram will be useful.

It doesn’t matter how big your TV is. If it’s an HDTV, the ratio of height to width will be 9/16. Here, I just have some constant in there (*a*) to get the actual size. Using the Pythagorean theorem, we can find the length of the diagonal based on the side lengths – you can see that parameter *a* survives. But this isn’t really right either. I want the area in terms of the quantity *s* (which I am using as the length of the diagonal). From this expression, I can get an expression for *a* in terms of *s*:

I probably shouldn’t have evaluated that square root since it looks like I will just square that value anyway. Now for the area (which I will label as *A*).

Let’s just check this really quickly. What if I have a 40 inch TV? It would have a height of 19.61 inches and a width of 34.86 inches. This would give it an area of 683.6 inch^{2}. Now, if I use the above formula with a diagonal of 40 inches, I get the same value.

Now for the plot of TV price vs. screen area.

That doesn’t look as linear as I would like it to be. Look at that 80 inch TV. It doesn’t fit very well with the linear function comparing area and price.

**More Data**

That doesn’t really improve anything. My last option is to forget about my first assumption. What if the different TV manufactures follow different models? Here is a plot of price vs. screen area for the Sharp, Target, and average Walmart TVs with separate linear fits.

This gives three different price functions for TVs in terms of screen area. (Yes, I know I left off some of the brands of TVs – but they mostly fit with the lower sized models so I left them off for a cleaner graph)

What does this tell us? First, if Walmart sold a TV with no screen (zero area) – it would cost you -$21.64. Yes, they would have to give you money. This could be my new job – buying zero-inch TVs and collecting the money. But really, does this make sense? Yes. It must mean that if they are basing the price on the area of the TV then they are subsidizing this price. Otherwise, your Target TV would cost 21 dollars more. And look at the Sharp TVs. They have a zero-inch price of almost $4,000. Of course, maybe this just indicates the sale price of these devices and not an actual price. What do I know? I am just making up economic stuff here.

**So, How Much? **

In this case, I have a question in my mind. What if there was a 100 inch TV? How much should that cost? What about a 200 inch TV? Of course, these are the diagonal measurements and not the area. So, let me re-write the price function in terms of diagonal size (*s*) instead of the area.

Now that the price is a function of size, I can just plop in the size to get a price. With this, a 100 inch TV should cost $10,706. A 200 inch TV would cost $54,806 dollars. That’s a serious TV. Now, this is just using the Sharp price function since I have data for larger TVs from them. What if Walmart made a 200 inch TV with their same pricing model? It would cost just $4,888. Not too bad.

One more thing. What if you want a free TV? How big would a Sharp TV be if it was free? Here, I can put in a price of zero dollars and then solve for the size.

That’s bigger than the TV I have right now. OK Sharp, I will take my free 52 inch TV. I will even pay for the shipping. Just send me an email and I will give you my address.

Oh – someone is probably going to ask if they can use my data. Sure. Here it is in a Google Docs spreadsheet. Have with it.

*Source*: Dot Physics