Additional Math Pages & Resources

Friday, September 17, 2010

After some reflection on visibility ...

It's still foggy this morning, so we might as well continue on the theme of visibility. Yesterday I said we would calculate the optimum eyeball height to see reflective clothing. I think the angle specified by 3M will equal the height of the eyes of a driver,  a few feet above the film and the headlights.

The 3M Scotchlite™literature says the optimum viewing angle is 0.2 degrees at 800 feet.

Let's make a right triangle whose adjacent side is 800 feet and whose angles are .2, 89.8, and 90 degrees. Remember that the sum of the angles of any triangle equal 180 degrees.

OK, there's our triangle. A is the height of the eyeballs, B is 800 feet, and C is the hypotenuse.

A few among us might recall the term Pythagorean theorem but can you remember what it says? Something about A² + B² = C² isn't it? Yes, but the real value is if you know a couple of the sides and angles of a triangle, you can discover the rest of them. We know all 3 angles and 1 side, so we can learn the lengths of the other sides.

This slightly above elementary math, but let's go ahead and calculate the height of A anyway. If you recall the formulas you can do this manually with a pencil, or a calculator (or go here to use a triangle side/angle calculator) then come back and check your answer against mine.

I get the height of A as 2.8 feet which is about 34-35 inches.

I needed someone to measure, so I drafted Darcie who does our order processing. We went out and measured her car's headlight height, and the difference in height from her headlights to her eyes when (1) standing up and (2) seated in the car.

1. Darcie is 5' 7" tall and her eyes are 37-38" above the headlights (depending on shoes!)
2. When seated in the car, her eyes are 48" from the ground; 24" above the headlights

What does all this measuring tell us? Reflective film works best when people are wearing it are within 200-300 yards from an oncoming vehicle. The majority of the light is reflected straight back at the driver, and the long range allows plenty of room to make decisions about turning or stopping.

The safety value of this kind of night-time attire was reinforced last night, when  3 girls walked out across the road in front of me in the middle of a very dark block. They had on regular clothing and were nearly invisible at less than 100 feet. They could see us coming but we almost didn't see them!

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