Expert: Paul Klarreich - 12/4/2006

Question
I've got a couple optimization problems that I am having trouble with.  
1) A sports complex is to be built in the form of a rectangular field with two semicircular areas at each end.  If the border of the entire complex is to be a running track 1256 meters long (inside edge of track), what should the dimensions of the complex  be so that the area of the rectangular field is a maximum?
 For my primary equation i used Area = pi*r^2 + xy (xy being the length and width of the rectangle, respectively)
My secondary equation is 1256 = 2x + 2y + 2pi*r
First of all, I'm wondering if my equations are correct, and if so, how to make the primary equation a function of only one variable.

2) At 3 pm an oil tanker traveling west at 15 knots pases the same point as a luxury liner that arrived at the same spot at 2 pm while traveling north at 25 knots.  At what time were the ships closest together? Hint: The primary equation can be set up as a function of one varaible so no secondary is needed.

I'm rather stuck on this one.  I drew out a picture and I think I need to minimize the distance, but I'm not sure and I don't even know how to begin :-/

Answer
Questioner:  Giselle
Category:  Calculus
 
Subject:  Optimization Problems
Question:  I've got a couple optimization problems that I am having trouble with.  
1) A sports complex is to be built in the form of a rectangular field with two semicircular areas at each end.  If the border of the entire complex is to be a running track 1256 meters long (inside edge of track), what should the dimensions of the complex be so that the area of the rectangular field is a maximum?
For my primary equation I used Area = pi*r^2 + xy (xy being the length and width of the rectangle, respectively)
My secondary equation is 1256 = 2x + 2y + 2pi*r
First of all, I'm wondering if my equations are correct, and if so, how to make the primary equation a function of only one variable.

2) At 3 pm an oil tanker traveling west at 15 knots pases the same point as a luxury liner that arrived at the same spot at 2 pm while traveling north at 25 knots.  At what time were the ships closest together? Hint: The primary equation can be set up as a function of one varaible so no secondary is needed.

I'm rather stuck on this one.  I drew out a picture and I think I need to minimize the distance, but I'm not sure and I don't even know how to begin :-/
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1) A sports complex is to be built in the form of a rectangular field with two semicircular areas at each end.  If the border of the entire complex is to be a running track 1256 meters long (inside edge of track), what should the dimensions of the complex be so that the area of the rectangular field is a maximum?

For my primary equation i used Area = pi*r^2 + xy (xy being the length and width of the rectangle, respectively)
My secondary equation is 1256 = 2x + 2y + 2pi*r
First of all, I'm wondering if my equations are correct, and if so, how to make the primary equation a function of only one variable.

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Yes, your TOTAL area is correct; it's the rectangle which is 'x' by 'y' plus the circle with radius 'r'.  HOWEVER,

A. the height of the rectangle, 'y' is the same as 2r.
B. you are really interested in the area of the rectangle only.
C. the track is twice the length (2x) plus a circumference.  So I think you have:

2x + 2 pi r = 1256, which you will solve for x:

x + pi r = 628
x = 628 - pi r

Now to maximize the rectangle: (You did say you wanted to maximize the rectangle, not the entire enclosed area, right?)

A = xy = 2xr
A = 2(628 - pi r)r

A = 1256r - 2pi r^2

Now you can do your stuff.

A' = 1256 - 4pi r
Set that equal to zero and solve:

1256 - 4pi r = 0
r = 314/pi
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2) At 3 pm an oil tanker traveling west at 15 knots passes the same point

>> which I will call the origin.

as a liner that arrived at the same spot at 2 pm

>> Whew! Remind me to stay on land.

while traveling north at 25 knots.  At what time were the ships closest together?   Hint: The primary equation can be set up as a function of one variable so no secondary is needed.
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I suppose you can use t (time) as your variable.  The question does say 'at what time'.  But it is useful to write:

x = distance of tanker from origin, after 3 PM
y = distance of liner from origin, after 3 PM
z = distance between boats after 3 PM.

Now  x^2 + x^2 = z^2, and:

x = 15t
y = 25(1 + t), because it traveled an extra hour before 3 PM.

So z^2 = 15t^2 + 25(1 + t)^2

Of course, z = sqrt(..) but we don't have to mess around with the square root.  If we minimize z^2, we will be minimizing z.

D(z^2) = 30 t + 50(1 + t)

30t + 50 + 50t = 0

80t = - 50
t = -5/8

Yes, a negative answer.  So the ships were nearest together at t = -5/8, meaning some time BEFORE they crossed the origin.