@x.in_hype
That reflection formula was unexpected 😮😮😊
@papa-i6b
I never knew people didn’t know you were supposed to reflect the triangle like that. We did a few problems like this in geometry
@geralynpinto5971
Very interesting. I too did not imagine a reflected point. How ingenious! I've learnt something new. Grateful for that. Thank you.
@otaku4ever2222
I expected to use the similar triangles feature but your way of modifying it was legendary.
@harshalshub5532
Same solution stands if we use the analogy of Fermat's principle of quickest path of light.
@RexxSchneider
Consider Northwestville, which is exactly 8 miles due North of Westville. The pipe length from either town would be the same. But now we can see that the shortest distance from Northwestville to Eastville is a straight line, which is the hypotenuse of a right triangle whose other sides are 10 miles (East-West) and 11 miles (North-South). The length of that line is then √(10^2 + 11^2) = 14.87 miles. And that's the minimum length of pipe. The river divides that triangle into two similar triangles in the ratio 7:11 (from the North-South side). So the length of the river from the point due North of Eastville to the power station will be 7/11 of 10 miles = 6.36 miles. The length of the river from the power station to a point due North of Westville is 10 - 6.36 miles = 3.64 miles. No calculus needed, no algebra needed.
@bjorntorlarsson
Build the pipeline perpendicularly from the canal to Westville, and from Westville to Eastville. The length is then 4+sqrt[10^2+(7-4)^2] =4+sqrt[109] ~= 14.440. Instead of here proposed sqrt[10^2+(7+4)^2] = sqrt[221] ~= 14.866. Saving almost 3%.
@NCR_Mapper
My way of finding the pipe length: In the 4:29 part, we can see that we can make an imaginary triangle. To find the height, we need to add the height of the two triangles which is 4 + 7 = 11. And from the beginning, we know that the length of the river is 10 which is gonna be our base for our imaginary triangle. Now using the pythagorean theorem, we can substitute the given. 11² + 10² = the length of the pipe squared. Solving it: 11²= 121 10²= 100 So: 121 + 100 = 221 Therefore the length of the pipe is the square root of 221 Which is approximately 14.87 (Please understand im just in 6th grade, my explanation skills are not so great.)
@roninkegawa1804
Such an elegant solution.
@DAANSlayerYT
I found it by simply inverting it(mirroring it to the opposite side. Drawing a straight line(shortest distance) and ended up with 14.8 miles as total length of pipe
@kushagrak1644
I used the reflection of light concept. Good problem.
@SkibidiMogger
I had a similar problem using the same solution on a geometry assignment. Instead it used speakers and wire length. The task was to find where on a cabinet would speak b being placed at result in the least wire length. Speaker a was 6 feet in the air, and placed at (0,0)speaker b 4 feet in the air and the cabinet 8 feet long. The answers was placing speaker b at (4.8,0) (if I remember correctly)
@J7Handle
I used a calculus line of reasoning to establish the similarity of the triangles. Sliding the point of the water station along the river by dx adds sin(X)*dx - sin(Y)*dx to the pipeline length, where X is the internal angle at Westville and Y is the angle at Eastville. We want a point where the total value added is zero, so sinX = sinY, and we want all points to the right and left of this point to add positive length to the pipeline, so when we go to the right of our point, sinX > sinY, and when we go to the left, sinX < sinY. These conditions are simply satisfied by X = Y, and we get similar triangles.
@crackwitz
Reflection. Light takes the shortest path. Flip one triangle up. Straight line. X=4/11*10
@jeffweber8244
14.87 miles total length.
@coderhub-tech7942
Is the solution not √221? Im pretty sure that the question asked for the length of the pipe and not the length of the river segment
@patrickkeller2193
Or you can simply use the law of intersecting lines. If the vertical segments have a ratio of 4/11 to 7/11 then the horizontal segments are 4/11*10 and 7/11*10. (using the mirrored diagram) And you can get the total pipe length with Pythagoras.
@MalfunctionNeo
Simplest option... construction, draw the setup on a page, copy one of the city's across the river, and draw a line from this new point to the other city., where the line crosses the river is the shortest, no math required.
@pauldowney9292
The pipe is a string, the station point is a slide... tighten the string to minimize it's length. You see the two angles are equal and the triangles are similar.
@mjorozco3786