emergent math

Feels like there’s a similarity (and a lot of other stuff) type problem in here.

Artifact

From CNET:

 In a perfect videophile world, you’d want to sit no closer than 1.5 times the screen’s diagonal measurement, and no farther than twice that measurement to the TV. For example, for a 50-inch TV, you’d sit between 75 and 100 inches (6.25 and 8.3 feet) from the screen. Many people are more comfortable sitting farther back than that, but of course the farther away you sit from a TV, the less immersive feeling it provides.

I’m wondering if you could pair this with Tim’s TV 3 Act problem. Perhaps even Brian’s Holiday Shopping problem. There’s honestly a lot of stuff going on here from CNET: proportion, distance, maybe even a system of equations or linear programming problem (what with the upper and lower bounds suggested above, then toss in cost constraints).

Update (6/12/17): CNET has apparently redirected the original article to a generic TV buying guide, so the above text is no longer viable. However, here’s something from The Home Cinema Guide.

A good rule of thumb is that the ideal viewing distance for a flat screen HDTV is between 1.5 and 3 times the diagonal size of the screen – and we can use this to calculate both approaches.

Still, the work for the rest of this article reflects CNET’s original viewing recommendations.

Guiding Questions

• How big a TV should I buy based on the above guidelines and my particular living room?
• Could we develop a mathematical model to illustrate these guidelines? With, like, variables and stuff?
• Alternatively, how could I set up my living room in order to fit the kind of TV I purchased?

Suggested Activities

• Have students develop a model (or “rules” to follow) to express the above recommendation mathematically. (This one’s partially answered below)
• Students could optimize viewing experience given a floorplan and a TV.
• A Consumer Reports-ish type TV buying guide? We’re veering here…

Attempted Solution

So the initial model for the constraints listed by CNET aren’t terribly complex.

Constraint 1) “you’d want to sit no closer than 1.5 times the screen’s diagonal measurement”

Constraint 2) “no farther than twice that measurement to the TV”

So lower bound: d>1.5x ; upper bound: d<2x ; and there you have it.

Surely we could ramp up the complexity of the problem with some of the above floorplanning activities and additional cost constraints. How would you modify this situation to serve our mathematical purpose here?