submarine A starts at (1000x,0y,0z) goes towards (0x,100y,1000z). It travels 72km/hour = 20m/sec. If submarine B travels at the exact same speed, what should its starting point be to reach (0x,0y,1000z) at the exact same time submarine A reaches (0x,0y,1000z)?
If A is going from (1000,0,0) to (0,100,1000) it will never reach (0,0,1000).
Anyway, when you decide how far A has to go, divide that distance by 72 to get the time needed.
Since you don't say how far B has to travel, it's tough to say how fast it needs to go ...
Steve,
You are right that A will never reach (0,0,1000). It was a typo, I meant to say (0,100,1000). But the question is not to find out how fast B needs to go. Ive already said its going the exact same speed as A. 20m/sec. Im trying to find out what should the starting point for B be such that if A and B are travelling 20m/sec, B will reach (0,0,1000) when A reaches (0,100,1000).
Oh, yeah. But the starting point can be anywhere on a sphere centered at B's destination.
To determine the starting point of submarine B, we need to consider the time it takes for submarine A to reach the point (0x, 0y, 1000z) at a speed of 20 m/s.
The distances between the starting point (1000x, 0y, 0z) and the ending point (0x, 0y, 1000z) in each dimension are:
- Δx = 1000x - 0x = 1000x (in the x-direction)
- Δy = 0y - 100y = -100y (in the y-direction)
- Δz = 0z - 1000z = -1000z (in the z-direction)
We can use the formula:
Time = Distance / Speed
Time taken by submarine A can be calculated as:
Time_A = sqrt((Δx)^2 + (Δy)^2 + (Δz)^2) / 20
Since submarine B is also traveling at the same speed, it would take the same amount of time to reach the ending point as submarine A. Therefore, the starting point of submarine B should be such that it takes the same amount of time to reach the ending point.
Let's assume submarine B starts at coordinates (Bx, By, Bz). Then, the distances between the starting point (Bx, By, Bz) and the ending point (0x, 0y, 1000z) would be:
- Δx' = 0x - Bx = -Bx (in the x-direction)
- Δy' = 0y - By = -By (in the y-direction)
- Δz' = 1000z - Bz = 1000z - Bz (in the z-direction)
We can use the same formula to calculate the time taken by submarine B to reach the ending point:
Time_B = sqrt((Δx')^2 + (Δy')^2 + (Δz')^2) / 20
For the two submarines to reach the ending point at the same time:
Time_A = Time_B
Therefore, we have:
sqrt((Δx)^2 + (Δy)^2 + (Δz)^2) / 20 = sqrt((Δx')^2 + (Δy')^2 + (Δz')^2) / 20
Simplifying further, we get:
sqrt((Δx)^2 + (Δy)^2 + (Δz)^2) = sqrt((Δx')^2 + (Δy')^2 + (Δz')^2)
Taking the square of both sides:
(Δx)^2 + (Δy)^2 + (Δz)^2 = (Δx')^2 + (Δy')^2 + (Δz')^2
Substituting the values of Δx, Δy, Δz, Δx', Δy', Δz', we have:
(1000x)^2 + (-100y)^2 + (-1000z)^2 = (-Bx)^2 + (-By)^2 + (1000z - Bz)^2
Simplifying further, we get:
1000^2x^2 + (-100)^2y^2 + (-1000)^2z^2 = Bx^2 + By^2 + (1000z - Bz)^2
Solving this equation will give us the values of starting coordinates (Bx, By, Bz) for submarine B to reach the ending point at the same time as submarine A.