Yacht A has initial position (-10, 4) and has velocity vector.
Yacht B has initial position (3, -13) and has velocity vector.
1. Explain why the position of each yacht at time t is given by
rA = + t and rB = + t
- For vector equations, the form is =. ( refers to an initial position and refers to a direction vector.)
- Therefore, a vector equation for Yacht A can be written as + t.
- A vector equation for Yacht B can be written as + t.
3. The position of B relative to A () is
rB – rA = + t, which in coordinates will be.
4. The formula for finding the distance is:
d =
Therefore, d2 = 169 - 78t + 9t2 + 289 - 136t + 16t2
= 25t2 - 214t + 458
5. d2 is a minimum when t = 4.28
To find the minimum of d2, I set the derivative equal to 0.
So, 50t - 214 = 0. Thus t = = 4.28.
6. The time when d is to be a minimum is the same time as when d2 is a minimum, so the closest approach occurs at t = 4.28. So, if I put t = 4.28 into the expression for d is:
d =
=
=
=
= 0.2 miles
