[Prof. Templeton]

Professor Templeton was staying at a cabin in the remote countryside and decided to go out horseback riding. After riding all day he found himself 8 km West of his cabin and 4 km North, but his horse needed fresh water before he could return. In his travels he had come across a river that ran due East to West, 6 km North of his current position. If if will take 15 minutes to water the horse and his horse can travel at a constant speed of 10 km per hour when thirsty and 12 km per hour when watered, what's the shortest time it will take Prof. T to return to his cabin?

I know only Chuck Norris take lead a horse to water and make him drink, but for this puzzle lets assume it will take 15 minutes.

[Guest]

Professor Templeton was staying at a cabin in the remote countryside and decided to go out horseback riding. After riding all day he found himself 8 km West of his cabin and 4 km North, but his horse needed fresh water before he could return. In his travels he had come across a river that ran due East to West, 6 km North of his current position. If if will take 15 minutes to water the horse and his horse can travel at a constant speed of 10 km per hour when thirsty and 12 km per hour when watered, what's the shortest time it will take Prof. T to return to his cabin?

I know only Chuck Norris take lead a horse to water and make him drink, but for this puzzle lets assume it will take 15 minutes.

36 minutes from current location to stream

15 minutes at the stream

67 minutes back to cabin

total time, 118 minutes.

[Prof. Templeton]

36 minutes from current location to stream

15 minutes at the stream

67 minutes back to cabin

total time, 118 minutes.

That would be a possible route, but is it the shortest?

[Guest]

That would be a possible route, but is it the shortest?

approx. 54 minutes, why water the horse, right?

[Guest]

approx. 54 minutes, why water the horse, right?

I'm pretty sure a dehydrated horse slows you down... It might die. Of course, you don't seem to care about animal rights. <_<

[Guest]

36 minutes from current location to stream

15 minutes at the stream

64.05 minutes back to cabin

total time, 115.1 minutes.

This is taking straight shot to river, due north and then making a bee line for cabin from there which would be 12.81 km

[Guest]

My math skills aren't that great but here is my guess

6 miles @ 10kph = 36

water horse = 15

10 miles @12kph = 36

total minutes = 87 minutes

[Guest]

My math skills aren't that great but here is my guess

6 miles @ 10kph = 36

water horse = 15

10 miles @12kph = 36

total minutes = 87 minutes

The return trip is > 10km. He is 10 km north and 8km west so he is (10^2 + 8^2)^1/2 km from cabin.

[Guest]

The Shortest distance between two points is a straight line, he is 4 miles north of his cabin and has to go another 6 miles north to water the horse. That means Due South it is 10 miles back to his cabin.

[Guest]

The Shortest distance between two points is a straight line, he is 4 miles north of his cabin and has to go another 6 miles north to water the horse. That means Due South it is 10 miles back to his cabin.

The shortest distance is indeed a straight line, but you are neglecting that he is also 8 km west of his cabin to begin with.

Sorry this is so crude, but it should get the point across.

[Guest]

The shortest distance is indeed a straight line, but you are neglecting that he is also 8 km west of his cabin to begin with.

Sorry this is so crude, but it should get the point across.

Okay so by combining your drawing and my conclusion, I had to add three miles to my assumption and now change my answer.

129 minutes

[Guest]

let s(1) = the distance the horse travel before it was watered.

s(2) = the distance the horse travel after it was watered.

since t = s / v

so total time, T = t(1) + 15min + t(2)

T = [s(1)*60]/ 10 + [s(2)*60]/ 12 + 15 min

= 6s(1) + 5s(2) + 15 (all in minutes)

the shortest T taken is when s(1) is minimum and hence, it goest straight to the north s(1) = 6km

then s(2) become square root of (8²+10²) = 12.806km

hence T = 6(6) + 5(12.806) + 15

T = 115.03 minutes. ?

[bonanova]

111.0589251 minutes.

Spoiler for Here's how:

w = the distance West of the cabin on the river where the horse waters.

Dt = distance traveled while thirsty [at 10km/hr, or 6 min/km]

Dw = distance traveled while watered [at 12 km/hr, or 5 min/km]

Return time is

[1] Tr = [6 Dt + 15 + 5 Dw] minutes

Minimize Tr with respect to w.

[2] Dt = [(8-w)² + 6²]^.5 - on the way to the river

[3] Dw = [w² + 10²]^.5 - from the river back to the cabin

It's clear that going straight north to the river [w=8] and going straight south from the river to the cabin [w=0] take longer than needed.

So somewhere between 8 and 0 w has a value w_min that minimizes Tr.

The roughest approximation for w_min is 4 km [halfway], but that's too low because PT starts north of the house

If the speeds to and from the river were the same, w_min would simply be 5 km - the route of shortest distance.

But since it's [slightly] better to travel a little farther watered, w_min will be somewhat greater than [west of] 5.

Here's what happens

• plug [2] and [3] into [1] and differentiate with respect to w.
  Use the chain rule for differentiating: if f(x) = f[g(x)], first set u=g(x); then df/dx = df/du x dy/dx.
  
• Set dTr/dw = 0
  
• Solve for w, getting w_min = 5.40889... km west of cabin
  
• evaluate [1]-[3] to get Tr(w_min) = 111.0589 ... minutes
  

Re Chuck Norris, [spoiler=You can lead a horse to drink ]But even Chuck can't make him water.

[Yoruichi-san]

96 minutes Edit: nvm...forgot the watering time...that gives me 111...(same as bonanova)

Edited December 5, 2008 by Yoruichi-san

[Guest]

Your error is assuming when s(1) is minimum, then T will be also minimum.

If you calculate in a math way, he shouldn't go straight to north, but 2.82 km. east of it.

As I 've forgotten the derivation of sqrt's, I reached result of 111 minutes by trying 2,3,4 for 2.82 value, and giving 3 km. to east travel, I got 111.

Thanks to Bonanova, as he reminded me derivations, and I managed to get the result without try and error.

Really, this is a math problem and it isn't possible to get it without derivations.

let s(1) = the distance the horse travel before it was watered.

s(2) = the distance the horse travel after it was watered.

since t = s / v

so total time, T = t(1) + 15min + t(2)

T = [s(1)*60]/ 10 + [s(2)*60]/ 12 + 15 min

= 6s(1) + 5s(2) + 15 (all in minutes)

the shortest T taken is when s(1) is minimum and hence, it goest straight to the north s(1) = 6km

then s(2) become square root of (8²+10²) = 12.806km

hence T = 6(6) + 5(12.806) + 15

T = 115.03 minutes. ?

[Prime]

I say... Hasn't Professor any water at home? Per Bonanova's calculations, the distance they traveled from their stop to water is only slightly shorter than the distance from that rest point to home. (It's something like 7 km vs. 9 km.) And if the horse refuses to go in any other direction, but towards the water, wouldn't it also insist on taking the shortest path to the river?

[Guest]

It started to heavily down pour and the Professor used his shirt to collect water to quench the horses thirst, then they returned home in a striaght line.

[Prof. Templeton]

It started to heavily down pour and the Professor used his shirt to collect water to quench the horses thirst, then they returned home in a striaght line.

Actually Prof. T used

Ten Gallon Hat