I love it when a common textbook problem comes alive. An employee of a Nebraska county road crew had to cut a culvert in two pieces so he could weld it back together and make a 90 degree angle with the pipe. They have a device called a master marker made by Flange Wizard that can be used to mark the angle needed, so that a torch can be used to cut the pipe. The problem with this is that the pipe they have(37 inch diameter)
was too big to use with the Flange Wizard they had on hand and to purchase a bigger flange wizard was out of the question. The question becomes then how do we create a mark on that pipe at 45 degrees with materials available. Before you read my solution, put some thought into this. How would you answer the question? My solution is not the only way. There are many methods for getting this done. How could you put this into your classroom to teach others? Organize it in acts like Dan Meyer or just throw the problem on the board and see what happens? I usually prefer the second scenario, but you have to be ok with some chaos.
The solution I come up with was to cut an ellipse out of a plane (cardboard) and slide the plane onto the pipe. I first found the major and minor axis of the ellipse. The minor axis is the radius of the pipe, in this case 18.5 inches. The major axis can be found by the hypotenuse of a 45-45-90 triangle with legs 37 inches. By pythagorean theorem the hypotenuse of this triangle is 37√2. The two red dotted lines represent a side view of the pipe. The blue line represents the cut that needs made and the long axis of the ellipse. The major axis of the ellipse is that number cut in half. This yields the equation of the ellipse as:
Once you have that equation it is possible to find the distance the foci of the ellipse are from the center of the ellipse along the major axis by taking the square root of the absolute value of the difference of the denominators in the equation. This results is a distance from the center of the ellipse to the foci of 18.5 inches. Here the fun begins. Once you measure from the center to the two foci it is possible to tack down a stretched piece of string between those two points and trace out the elliptical shape on the cardboard. Cut this shape out and then slide it over the pipe. The crew put wood slats on the cardboard in order to keep it from bending.
I missed the boat with my students on this one. I am in the heat of battle teaching the Nebraska State Standards. I think it would have been fun to put them in groups with the problem and see what they came up with, had them present their proposals to the actual people doing the work, and let the people doing the work decide on the best proposal. I just didn’t feel like I could sacrifice the time. But I will put this idea on the shelf and maybe next year with better organization fit it into the curriculum some where. Later this spring this group also needs to cut similar pipe at different angles, so maybe we will have more time. So there is already an extension of the problem built in. I think they need 22 degree angles cut next time so they can send water down a 40% grade.