Real life engineering word problem..?

Question

A few colleagues and I disagree about the answer to the following problem:
A shower door hinge should sit at 135 degrees. The door piece is attached to another piece of glass that is fastened to the wall at a 90 degree angle. The opening side of the door also rests against another piece of glass that is fastened at a 90 degree angle to the wall.
We inspected some samples that measured up to 139 degrees (4 degrees too wide). If the door is 1 meter in length, how far away from the second glass sheet would the opening end of the door be sitting (while the door is closed).
How did you solve it?

Thanks!!

Answer 1

If I understand your question, you want to make a pentagon with angles 90,90,90,135,135. You're asking how wide a gap remains if the door hooks in at 139 instead of 135.

For a first estimate, I decided to figure out the arclength that results when a 1 meter door is swept through a 4 degree angle. The actual distance, being a straight line, will be slightly shorter than this curve. I took the circumference of a 1m radius circle (2 pi meters) and multiplied by 4/360 = 1/90. I end up with a distance of pi/45 meters, or roughly 6.981 cm.

Second attempt: Law of Cosines. In my mind, there is a triangle with three sides: the ideal door (at 135 degrees), the actual door (at 139 degrees), and the distance between the two (an unknown distance x). The angle between the two possible doors is 4 degrees, so Law of Cosines says x^2 = 1 + 1 - 2(1)(1)Cos(4). Solving for x, I found that the unknown distance was 6.980 cm.

Note that this distance cannot be bridged by merely extending one of the 90 degree panes, because the door will then be a shade too long for a nice union. Assuming that the walls of the shower lie along the positive x- and y-axes (so that the door itself lies entirely within the 1st quadrant), I estimate that the lip of the door will move 5cm up and 5cm right. You might extend a pane by 5cm and move the same pane 5cm away from the wall it parallels.

Hope this helped!

Answer 2

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