SOLUTION: Two pathways meet at 30° to each other. One pathway has lighting and the other does not. The distance between successive lights on the lighted pathway is 5 metres. Each ligh

Question 1162480: Two pathways meet at 30° to each other.
One pathway has lighting and the other does not.
The distance between successive lights on the lighted pathway is 5 metres.
Each light has a range of effective illumination of 6 metres.
What length of the pathway without lights is illuminated by the pathway with lights? (Round your answer to the nearest tenth, if necessary.)
I drew a diagram of a 30-60-90 triangle, but I'm not sure how to find out up to which point the unlighted path is illuminated. Thank you!!

Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

You need the distance between the two blue points. Find the

-coordinate of the one on the positive side of the

-axis by solving:

Note that the offset for the independent variable on the circle function is actually arbitrary because the two ends of the illuminated part of the unlit path will move proportionately as the actual placement of the lamps on the other path shift. The particular configuration shown was chosen to simplify computation.

Note that the ratio of the long leg of a 30-60-90 right triangle is in proportion

, so once you have the

-coordinate of the circle/line intersection on the right side, multiply by

to get the measure of the slant path from the origin to the blue point intersection. Then just multiply by 2 to get the entire length.

John

My calculator said it, I believe it, that settles it

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