I have homework that I have to solve, and the my question is:
A cube has side length of 2 cm. One ant starts at corner A and travels to corner B.
Another ant starts at corner C and travels to corner D. The two ants leave at the same time and travel at the same speed. Find the shortest distance between the two ants.
and my solution:
I don't know if my answer is correct, is it?
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O-BL
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The lines are not paralell (even thogh you say they are) – 5xum Oct 13 '17 at 07:54
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1Welcome to MSE. Please type your questions, instead of posting images. – José Carlos Santos Oct 13 '17 at 07:56
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It shoud be $+12$ instead of $+8$. – Oct 13 '17 at 08:01
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Thanks @JoséCarlosSantos for your advice, I have edited the question, could you check it out. – O-BL Oct 13 '17 at 08:01
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@O-BL I've edited your question, but it would be better if you would type your solution. – José Carlos Santos Oct 13 '17 at 08:12
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@JohnMa Thank you, I have edited my solution could you reload? and check if it correct? and I will be thankful! :) – O-BL Oct 13 '17 at 08:22
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@JoséCarlosSantos Thank you, actually I don't know how to format the formulas, so I think i would keep an image. – O-BL Oct 13 '17 at 08:23
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@O-BL Please read the MathJax tutorial. – José Carlos Santos Oct 13 '17 at 08:24
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Check your expressions at $t=0$ and $t=1$. – Oct 13 '17 at 08:25
1 Answers
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The second ant is at $(2,2,-2t)$.
The squared distance is
$$(2t-2)^2+(-2)^2+(2t)^2=8(t^2-t+1).$$
By completing the square, the minimum is achieved at $t=\dfrac12$, when the ants are at the midpoints, and the distance is $\sqrt6$.
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By the way, it is easier/safer to write the coordinates of the ants by simple inspection than by error-prone computation of the line coefficients. – Oct 13 '17 at 08:32
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Could you explain a little further? I believe that the second ant starts at C, so at t = 0 the z-coords should be = 2, and you say it is just zero! – O-BL Oct 13 '17 at 08:38
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let's see; hmm, so the second ant z equation is not correct since C is (2,2,0), I see; and it should be as you said in the answer, thank you a lot, that helped. – O-BL Oct 13 '17 at 08:56