Optimization Woes....

[tangents]

Greetings, My AP Calculus AB class is doign optimization but the problem is the teacher is zooming past this topic in 3 days. We get to about 2 questions a day because it takes our class such a long time to understand whats going on. Anyway i sort of get that we have to do in that we are trying to maximize or minimize something. So anyway i am stuck on this problem and wish for some assitance in setting it up with the primary equation/secondary.

"At 1pm, ship A is 30 miles due south of ship B and is sailing north at a rate of 15 miles/hour. If ship B is sailing west at a rate of 10 miles/hour, find the time at which the distance d b/w the ships is minimal"

Ship B--------------------------------------port
- .
- - .
- - - .
- - - - - .
-- - - - . Ship A ( line should be longer_

That should be a slanted line forming a triangle lol

Anyway the only equtaions that come acros my mind are D=V/T and the distance formula. I am tring to minimize time/distance. an i think i only have one variable to work with so it cant be that difficult. I have to find my primary equation, get the derivative, find crit #'s do either first/second derivative test to see which is realtive min. Problem is I duno how to get primary equation. Any help will be apprecaite.

 

[galactus]

Using ol' Pythagoras, we must minimize D, the square of the distance between

ships. Let t=hours past 1 pm.

The distance ship A has sailed since 1 pm is \(\displaystyle 30-15t\)(since it was initially 30 miles away and 15t because it is travelling at 15mph). Let that be \(\displaystyle y^{2}\)

Let \(\displaystyle x^{2}\) be ship B's distance travelled since 1 pm, \(\displaystyle 10t\).

So we have \(\displaystyle D(t)=(30-15t)^{2}+(10t)^{2}\)

\(\displaystyle D'(t)=2(30-15t)(-15)+100(2t)=650t-900\)

\(\displaystyle 650t-900=0\)

\(\displaystyle t=\frac{18}{13}=1.38\)

in other words, 1.38 hours equals about 1 hour and 23 minutes. Therefore, the minimum distance will be at 2:23 PM.

Please check my work. Easy to err.

 

[tangents]

Thank you, know I see what you did and yes that answer matches the one in my textbook. Would you have any good tutorials maybe for optimization problems because its always the primary equation i never know how to figure out. The rest is all straightforward. Do you recommand me drawing diagrams for these problems on tests? and are these similar to related rates a well?