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Three Towns

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The towns of Alpha, Beta, and Gamma are equidistant from each other. If a car is three miles from Alpha and four miles from Beta, what is the maximum possible distance of the car from Gamma? Assume the land is flat.

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Posted · Report post

Don't have pencil and paper handy.

It occurs to me the towns are 1 mile apart and the car lies on a line joining alpha and beta.

Its distance from gamma is 3.5 miles along that line, say north, and the height of the triangle say to the east.

Apply Pythagoras.

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Posted (edited) · Report post

Don't have pencil and paper handy.

It occurs to me the towns are 1 mile apart and the car lies on a line joining alpha and beta.

Its distance from gamma is 3.5 miles along that line, say north, and the height of the triangle say to the east.

Apply Pythagoras.

They don't have to be 1 mile apart and i got 3.61 when they were 1 mile apart

Edited by BMAD
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If the side of the equilateral triangle were 5, and the car formed a 3,4,5 triangle with Alpha and Beta, then the car would be 6.77 from Gamma.

It isn't as good to put the car on the line between alpha and beta, with a side of 7, as the car would be 6.01 from gamma.

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Posted · Report post

If the side of the equilateral triangle were 5, and the car formed a 3,4,5 triangle with Alpha and Beta, then the car would be 6.77 from Gamma.

It isn't as good to put the car on the line between alpha and beta, with a side of 7, as the car would be 6.01 from gamma.

That is a nice improvement CaptainEd but still even that isn't the maximum distance.

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I'm finding it hard to believe, so I'm second guessing myself that I messed up somewhere in the calculations, but the answer I got is 7. It occurs when the angle formed between Alpha,Car,Beta is 120 degrees and the distance between the towns is sqrt(37).

Edited by k-man
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Posted · Report post

I'm finding it hard to believe, so I'm second guessing myself that I messed up somewhere in the calculations, but the answer I got is 7. It occurs when the angle formed between Alpha,Car,Beta is 120 degrees and the distance between the towns is sqrt(37).

I found the distance between the towns to be the sqrt(36) or 6.

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Posted (edited) · Report post

...and sqrt(37) is the correct distance between the towns. For any

a and b, where a is the distance from the car to Alpha and b is the distance from the car to Beta, the maximum distance from the car to Gamma is a+b and is achieved when the angle formed by the lines connecting the car with Alpha and Beta is equal 120 degrees. Applying the law of cosines with 120 degree angle we get the distance between towns = sqrt( 32 + 42 - 2 * 3 * 4 * cos(pi/3) ) = sqrt(37).

post-9659-0-33333900-1361829371_thumb.pn

Edited by k-man
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Posted · Report post

...and sqrt(37) is the correct distance between the towns. For any

a and b, where a is the distance from the car to Alpha and b is the distance from the car to Beta, the maximum distance from the car to Gamma is a+b and is achieved when the angle formed by the lines connecting the car with Alpha and Beta is equal 120 degrees. Applying the law of cosines with 120 degree angle we get the distance between towns = sqrt( 32 + 42 - 2 * 3 * 4 * cos(pi/3) ) = sqrt(37).

attachicon.gifthree towns.png

I stand corrected. I didn't catch that i rounded :b

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Posted (edited) · Report post

gamma should be (3^1/2)3.5/2 miles apart if we draw two circle with center alpha and radius 3miles and beta with radius miles we see that gamma will see that they will form an equilateral triangle with side 3.5 iles applying pythagoras theorem we will get the maximum distance....

now i found out that i am wrong .....Happens!

Edited by Debasis
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Posted · Report post

...and sqrt(37) is the correct distance between the towns. For any

a and b, where a is the distance from the car to Alpha and b is the distance from the car to Beta, the maximum distance from the car to Gamma is a+b and is achieved when the angle formed by the lines connecting the car with Alpha and Beta is equal 120 degrees. Applying the law of cosines with 120 degree angle we get the distance between towns = sqrt( 32 + 42 - 2 * 3 * 4 * cos(pi/3) ) = sqrt(37).

attachicon.gifthree towns.png

hey man can you tell me how you worked out that the angle between alpha and beta should be 120 degrees .i just acant understand..Thanks

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...and sqrt(37) is the correct distance between the towns. For any

a and b, where a is the distance from the car to Alpha and b is the distance from the car to Beta, the maximum distance from the car to Gamma is a+b and is achieved when the angle formed by the lines connecting the car with Alpha and Beta is equal 120 degrees. Applying the law of cosines with 120 degree angle we get the distance between towns = sqrt( 32 + 42 - 2 * 3 * 4 * cos(pi/3) ) = sqrt(37).

attachicon.gifthree towns.png

hey man can you tell me how you worked out that the angle between alpha and beta should be 120 degrees .i just acant understand..Thanks

I knew someone would ask that question :)

It was a little messy and maybe not in the most efficient way, but...

I used the laws of cosines and sines to express the distance to Gamma as a function of the angle between Alpha and Beta. Then I to took a derivative of that function to find the local maxima and found it to be at 120 degrees. If I have more time later, I may post the complete solution

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Posted · Report post

...and sqrt(37) is the correct distance between the towns. For any

a and b, where a is the distance from the car to Alpha and b is the distance from the car to Beta, the maximum distance from the car to Gamma is a+b and is achieved when the angle formed by the lines connecting the car with Alpha and Beta is equal 120 degrees. Applying the law of cosines with 120 degree angle we get the distance between towns = sqrt( 32 + 42 - 2 * 3 * 4 * cos(pi/3) ) = sqrt(37).

attachicon.gifthree towns.png

hey man can you tell me how you worked out that the angle between alpha and beta should be 120 degrees .i just acant understand..Thanks

I knew someone would ask that question :)

It was a little messy and maybe not in the most efficient way, but...

I used the laws of cosines and sines to express the distance to Gamma as a function of the angle between Alpha and Beta. Then I to took a derivative of that function to find the local maxima and found it to be at 120 degrees. If I have more time later, I may post the complete solution

thanks!

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