Dog and Decking Quiz Puzzle Answer

At Quiz Master Shop Towers we have some decking that is five metres wide. At one end it is three metres deep and the other end it is two metres deep, making an irregular trapezium.

There is a wall along the five metre and three metre sides. 

A friend sometimes brings their dog and we sit with a drink putting the world to rights. 

However, we are a bit concerned about the dog falling off the decking. What is the longest piece of rope that we can use to tie the dog so it can just reach the edge, but not fall off? 

The path described by the end of the rope is an arc of a circle, and the longest possible rope will make an arc with the diagonal side as a tangent to the arc.

Assuming the rope is tied in the corner between the five metre and three metre sides, we have a right angled triangle with a three metre hypotenuse. 

Imagine another triangle that forms the part of the trapezium that isn't the two metre by five metre rectangle. This is a right angled triangle with a common angle with the first triangle - similar triangles. 

Tan of the smallest angle in the second triangle is 1/5, so the angle is Arctan (1/5) = 0.157079633 radians. 

Returning to the first triangle the length of the rope is 3 * Cos x = 3 * 0.98768834054 which is about 2.96 metres.


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