# Optomization problem using integrals

## Homework Statement

A boat leaves a dock at 2:00 pm and travels due south at a speed of 20 km/h. Another
boat has been heading due east at 15 km/h and reaches the same dock at 3:00 pm. At
what time were the two boats closest together?

3. The Attempt at a Solution [/b

We actually don't need to use integrals for this(and we haven't really learned how), but I want to.

So a couple of quick questions...
first, integrating 20 equals 20t+c correct?

and second, either the 15km/h or the 20km/h should be negative, and it doesn't matter which one, right?

If those two things are true, I should be all set, if not, I might have more questions. :p

Thanks a lot for the help guys.

Related Calculus and Beyond Homework Help News on Phys.org
Mark44
Mentor
You're overthinking this problem. Both boats are travelling at constant rates, so the distance they travel in t hours is going to be 15t km and 20t km, using the formula d = rt. That's all your integration has bought you, plus you have two constants of integration to worry about.

If you haven't drawn a picture, you should, and maybe two of them, one for the positions of the two boats at 2:00 and another for their positions at 3:00. You need an expression that represents the distance between the two boats, as a function of t, and that's what you have to minimize, using differentiation.

right, I have that position with the Pythagorean theorem. I have a^2+b^2=c^2. is a the position function of one boat and b the position function of the other boat? If that's the case, then I have c as a function of time if I integrate the velocities.

The integration constants aren't a problem, because I know the position of one boat at t=0 and the other at t=1 so I can solve for c pretty easily.

Then all I have to do is find the minimum of the Pythagorean function I created which looked to be 9/25.

I'd add more of the math I've done to make what I'm saying more clear, but with latex down... :/