Expert: Paul Klarreich - 12/17/2007

Question
QUESTION: http://en.allexperts.com/q/Calculus-2063/Application-Differentation.htm

i was looking at this problem, and im unsure as to why you divided line segment AB by 2, and arc BC by 2, and then just dismissed that when adding them together to find the total distance. is this a simple algebra rule im missing?

ANSWER: Hi, Sara,

I think I got it right.  I find this in the answer:

=======================================
Rowing time = AB/2 mph. = sqrt(x + 2)

Now the walking part, along the arc from B to C.  Use the arc-length formula, s = tr.
(remember, t is theta)
What's theta in terms of x?  cos t = x/2, so  t = arccos(x/2)

So arc BC = 2t, and walking time = arc BC/4 = t/2 = (1/2)arccos(x/2)

======= Divided arc BC by 4, not by 2. ============

Note that the objective was to minimize time, not distance.

But thank you, anyway.  I am always glad to see people reading my stuff.  And any time there is something in an answer you don't understand, you are right to call me on it.  I used to be perfect, now I'm just old.




---------- FOLLOW-UP ----------

QUESTION: Hi Paul, I'm not trying to revise you, I'm trying to see what you've done because I have this problem to do as well. I'm no genius, and I dont understand WHY you divided AB by 2 and arc BC by 4, can you help me out? :) thanks.

Answer
ANSWER: Hi, Sara,

I think I got it right.  I find this in the answer:

=======================================
Rowing time = AB/2 mph. = sqrt(x + 2)

Now the walking part, along the arc from B to C.  Use the arc-length formula, s = tr.
(remember, t is theta)
What's theta in terms of x?  cos t = x/2, so  t = arccos(x/2)

So arc BC = 2t, and walking time = arc BC/4 = t/2 = (1/2)arccos(x/2)

======= Divided arc BC by 4, not by 2. ============

---------- FOLLOW-UP ----------

QUESTION: Hi Paul, I'm not trying to revise you,

>> No problem with that.  Now, reviling -- that's a problem.(Just kidding, of course.)

I'm trying to see what you've done because I have this problem to do as well. I'm no genius, and I dont understand WHY you divided AB by 2 and arc BC by 4, can you help me out? :) thanks.
>> --------------------------------
Because the problem asked to minimize the time needed to cross the lake, we want to express Total Time in terms of x:

Basic principle:
         distance traveled
Time(to travel) = -----------------
         speed.
         distance rowed across lake = AB
Time(to row) = -------------------------------
         rowing speed = 2 mph

         distance rowed AROUND lake = arc BC
Time(to walk) = ----------------------------------
         walking speed = 4 mph


Then add those for total time.