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Problem 81 Hard Difficulty

Two tanks are participating in a battle simulation. Tank $ A $ is at point $ (325, 810, 561) $ and tank $ B $ is positioned at point $ (765, 675, 599) $.

(a) Find parametric equations for the line of sight between the tanks.
(b) If we divide the line of sight into 5 equal segments, the elevations of the terrain at the four intermediate points from tank $ A $ to tank $ B $ are $ 549, 566, 586 $, and $ 589 $. Can the tanks see each other?

Answer

(a) $x=325+440 t, y=810-135 t, z=561+38 t$
$0 \leq t \leq 1$
(b) No

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Video Transcript

for this problem here, um, were given are two points, so we can connect them. And this is going to give us our vector. So using our points and subtracting, we get that. Our victory is 7 65 minus 3 25 um, 6 75 minus 8, 10 and 5 99 minus 5. 61. So it's gonna give us 4. 40 negative. 1, 35 38. And then the position vector for the 0.32 or 3. 25 8 10 5. 61 is going to be 3. 25 8, 10 5, 61. So now what we have is sure Position vector being equal to 3. 25 8, 10, 5 61 plus t times this, plus two times that. And then when we solve it all out, we end up getting that X equals 3 25 plus 4. 40 t y is equal to 8. 10 minus 1. 35. T and Z is equal to 5. 61 plus 30 18. Then we go into the second part. Um, since it is given that the elevation of the terrain at the four intermediate points are 5 49 5 66 5. 86 In 5 89 What we want to do is take RZ components inside hostels the elevation. That's 5 61 plus 30 80. And we want to test it in our different values of t, which are gonna be 1/4. 2 Sorry. 1/5. 2/5. 3/5 4 5th. What we end up getting is 5. 68 6 5, 76.2 5, 83 8 5, 91 4. I mean, these are gonna be the elevations of the line of the site, from tank to tank. Be at those four intermediate points. Let me see. Since the third intermediate point from Tank A of the line of sight has an elevation that is lower. We see the terrain blocks the site and therefore Tank A and B will not see each other.