GeekDad Puzzle of the Week: Taking the Measure of the River -- Solution

This week, our intrepid NationalGeekographer needed your help in measuring the width of a river in her way as she travels the jungles of the world. Congratulations to David who submitted one of many correct answers. He will receive a $50 gift certificate from ThinkGeek! The rest of you can read to the end of this post to find your code for $10 off a $40 or more purchase. Puzzle and solution to follow.

Our geekographer occasionally has to travel through exotic landscapes to find and map the location of all of the geek parents in the world. This week, she’s come across a large river blocking her path. She has no means of crossing the river. All she has is a radio controlled helicopter onto which she can tie a guide line which she then uses for exact placement of one end of her self-contained bridge. Unfortunately, the range of her helicopter is rather limited and she doesn’t want to lose it to the river. The river seems to be just at the edge of the helicopter’s range, but she is unwilling to test it. How can she determine the width of the river? She has a measuring tape of 150 meters length and some antique surveying equipment. Remember, neither she nor any of her equipment can cross the river, and she doesn’t have any fancy equipment that will tell her the distance to the other side.

Having been trained on old style surveying equipment (called a transit) this is what I'd do:

For this case I'm assuming the helicopter can fly 150 metres.

At the point at which you wish to cross (pt. A), use the transit to sight an object across the river, ideally along the rivers edge. This spot across the river is point R.

Rotate the transit 180 degrees (away from the river), with the measuring tape drive a stake 150 metres from the transit along the line she is sighting. This spot is point Z.

Return to the transit and rotate the scope 90 degrees and drive a stake along this sight line. The distance doesn't matter but a length of about 150 metres is optimal (in this example). This spot is point X.

Stake below where the transit is (pt. A) then move the transit to point X.

At point X find the object across the river - point R - and measure the angle between point R and point A. This angle is theta.

Sight point A again and rotate the transit scope until you can view point Z. This angle is rho, if rho is greater than theta then the distance across the river is less than 150 metres. If it is less, than the distance is greater than 150 metres.

The above assumes that the R/C copter's range is 150 metres, if it was less I would measure less distance between A and Z. I.E. if the range was 75 metres then measure A - Z as 75 metres. If it was longer than 150 metres then I'd measure 150m temporarily marke the spot, then add the extra distance to find pt. Z.

The distance between A and X should be about the same as A to Z, this would imply that the angle rho would be about 45 degrees and that maximizes the change in distance for a given change in angle.

This isn't perfect but you could do it with basic tools and you don't
need trig tables, a calculator or a slide rule.

In real life you'd use trigonometry to work out the distance with the distance A - R being equal to (distance A to X)*tan(theta)

Of course, you do not have to stay in the approximate plane of Earth's surface. I received several (correct) solutions that required climbing trees. I don't know if our geekographer is good at climbing trees, but she better return the theodolite in one piece. ;) There are, of course, many similar ways to get the answer, and several people gave good ways to approximate the distance without using any equipment at all.

Thanks for playing! Please come back next week for another puzzle! Don’t forget to take $10 off a $40 or more purchase from ThinkGeek by using the code, GEEKPUZZLER.