Optimal Rear-View Mirror Width Calculation for Improved Car Safety

[mustang]

The rear-view mirror of a car is so placed that its upper and lower edges are horizontal and its cener is at the same level as the center of the rear window. The driver's eye is also at this level, and the line of sight from his eye on the center of the mirror makes an angle of 30 degrees with the line joining the centers of the mirror and the window. The distance from his eye to the mirror is 2 feet, and that from the mirror to the window is 8 feet. What is the least width of the mirror is needed if the entire width (3 feet) of the rear window is to be seen?
Would I do 3/8=x/2?

 

[Chen]

A drawing might help here.

Since you know the red angle, you can also know where the driver seats in relation to the center of the mirror/window line (green line). Now you can find the length of the orange line (half the mirror length minus the green line), and you should alreayd know the length of the purple line (half the window length minus half the mirror length).

Now you have two similar triangles, with the purple and orange lines as their bases. You know the heights of the two triangles, so you know what the ratio between the purple line and orange line is. From there you find the length of the orange line, and once you know that you can find the length of the mirror itself.

We ignore the other side of the mirror, because if the driver can see the left side of the window, there's no doubt that he can see the right side of it as well.

The answer, if my rapid calculations are correct (and I wouldn't count on that), is 2.45 feet.

 

[Chen]

Forgot the drawing:
http://www.avinadav.com/problem.gif

Sorry I used colors instead of variable names, by the way... it's just easier to paint with colors.