# Thread: Confused on an Optimization problem

1. ## Confused on an Optimization problem

I've tried this problem but I don't seem to be getting anywhere. At 1:00 pm ship A is 30 miles due south of ship B and sailing north at 15 mi/hr. If ship B is sailing west at a rate of 10 mi/hr, find the time in which the distance d between the ships is minimal.I would just like to get clues on how to start it, but don't solve it for me.Thanks.

2. Originally Posted by Arturo_026
I've tried this problem but I don't seem to be getting anywhere. At 1:00 pm ship A is 30 miles due south of ship B and sailing north at 15 mi/hr. If ship B is sailing west at a rate of 10 mi/hr, find the time in which the distance d between the ships is minimal.I would just like to get clues on how to start it, but don't solve it for me.Thanks.
how about Pythagoras?

$D^2 = (10t)^2 + (30-15t)^2$

take the derivative w/r to time, set $\frac{dD}{dt} = 0$, and minimize

3. Originally Posted by Arturo_026
I've tried this problem but I don't seem to be getting anywhere. At 1:00 pm ship A is 30 miles due south of ship B and sailing north at 15 mi/hr. If ship B is sailing west at a rate of 10 mi/hr, find the time in which the distance d between the ships is minimal.I would just like to get clues on how to start it, but don't solve it for me.Thanks.

So let's create a position function for A, it starts 30 miles due south of b, and then moves 15 mi/h so let's claim that A starts at the point (0,-30) which means B is at (0,0)

so for A

x=0
y=15t-30

So when t=0, we're at (0,-30)

and for B

x=10t
y=0

So when t=0, we're at (0,0)

Now the distance between the ships at time $d=\sqrt{(0-10t)^2+(15t-30-0)^2}=\sqrt{100t^2+(15t-30)^2}$

But we can simply minimize d^2, so let's take the derivative of d^2

$
(d^2=100t^2+(15t-30)^2)
$

$=200t+2\cdot 15(15t-30)=0$

$200t+450t-900=0$

$t=\frac{900}{650}$

And i suppose you should check to make sure this really is a minimum, but it is

4. Thanks so much.I solved it and got the right answer. However I still have some question......did you guys have to assume that ship B was starting at the point where ship A was to arrive at t=2? (since it's not mentioned in the problem)............OR would the formula still work if ship B's starting position was any?
..................And why are the format of my postings coming all weird and crumpled?......

5. Originally Posted by Arturo_026
Thanks so much.I solved it and got the right answer. However I still have some question......did you guys have to assume that ship B was starting at the point where ship A was to arrive at t=2? (since it's not mentioned in the problem)............OR would the formula still work if ship B's starting position was any?
..................And why are the format of my postings coming all weird and crumpled?......
just used the fact that ship A started 30 miles south of B ... the fact that it takes 2 hrs to reach B's starting position is irrelevant.