Freedom physics
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Question: Winnie lives in a village in rural Africa which is marked as (2,3) on a map.

Each day she goes to a river which flows due East, she fills a bucket with water at R and takes it to her grandmother who lives in the nearby Village Q. Winnie wants to know where to fill the buckets so that she has the shortest distance to walk.

P is (2,3), Q is (8,1), the equation of the river is y=5, please may you tell me why the shortest route from P to the river and then to Q is P -> (4,5) -> Q? 😄
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RDKGames
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(Original post by Freedom physics)
Question: Winnie lives in a village in rural Africa which is marked as (2,3) on a map.

Each day she goes to a river which flows due East, she fills a bucket with water at R and takes it to her grandmother who lives in the nearby Village Q. Winnie wants to know where to fill the buckets so that she has the shortest distance to walk.

P is (2,3), Q is (8,1), the equation of the river is y=5, please may you tell me why the shortest route from P to the river and then to Q is P -> (4,5) -> Q? 😄
Say (x,5) is a point on the river where she SHOULD go in order to minimise distance.

Then the total distance to walk is
\begin{aligned} D & = \sqrt{(x-2)^2 + (5-3)^2} + \sqrt{(8-x)^2 + (1-5)^2} \\ & = \sqrt{x^2-4x+8} + \sqrt{x^2 - 16x + 80}

Now you want to minimise D. This means setting \dfrac{dD}{dx} = 0. This then yields that x=4.
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ghostwalker
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(Original post by Freedom physics)
...
RDKGames has given you one method, though there is no need to get into calculus to solve this.

From P, she has to go up to R.

If you reflect P in the line y=5, to P' say, you have effectively the same problem. P and P' are as far from the river as each other, and the distance to any point on the river is the same for each.

The problem now becomes what's the shortest distance from P' to Q, and it will automatically go via the river.
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RDKGames
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(Original post by ghostwalker)
RDKGames has given you one method, though there is no need to get into calculus to solve this.

From P, she has to go up to R.

If you reflect P in the line y=5, to P' say, you have effectively the same problem. P and P' are as far from the river as each other, and the distance to any point on the river is the same for each.

The problem now becomes what's the shortest distance from P' to Q, and it will automatically go via the river.
That's neat!
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Prasiortle
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(Original post by RDKGames)
That's neat!
Yes, we commonly use this reflection method in competitions as when you're under time pressure, it's much faster to reflect appropriately and then use "the shortest distance between two points is a straight line", rather than trying to do calculus on an expression involving a bunch of radicals. The method actually goes back to Heron of Alexandria (same guy who discovered Heron's Formula for the area of a triangle): see e.g. https://www.maa.org/press/periodical...stance-problem
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Freedom physics
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Thank you! I'll give the method a try now! 😄
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Freedom physics
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The book used the same method although it reflected Q instead, although I found the equation of the line connecting P' to Q and when you sub in y=5 into it it gives R as (5,5), please may you help me with this? 🙂
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Freedom physics
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Tbh I think the book's miscalculated something because I've just done it their wae and for exactly the same result!
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ghostwalker
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(Original post by Freedom physics)
The book used the same method although it reflected Q instead, although I found the equation of the line connecting P' to Q and when you sub in y=5 into it it gives R as (5,5), please may you help me with this? 🙂
You've made a slip somewhere, since the point should be (4,5).

Post your working. You should have P' is the point (2,7).
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Freedom physics
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Why would P' be (2,7), I thought it was (2,8) because P is three units away from y=5 so P'should be three units above it 🙂
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ghostwalker
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(Original post by Freedom physics)
Why would P' be (2,7), I thought it was (2,8) because P is three units away from y=5 so P'should be three units above it 🙂
P is the point (2,3) which is 2 units away from (2,5)!
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Freedom physics
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Oops I got them the wrong way around! 😂
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Freedom physics
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Thanks for pointing that out! And when I said them I referred to the coordinates of P! 😄
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ghostwalker
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(Original post by Freedom physics)
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All sorted now?
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Freedom physics
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Yeah, I'm not sure how that reply came out as that because I posted something completely different and never even typed a ... 😂
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PQE
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(Original post by Freedom physics)
Why would P' be (2,7), I thought it was (2,8) because P is three units away from y=5 so P'should be three units above it 🙂
P is 2 vertical units away and has the same x coordinate because the line is flat.
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