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Subject: "density question"     Previous Topic | Next Topic
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bob
guest
Feb-26-06, 10:13 AM (EST)
 
"density question"
 
   a straight road goes through the center of a circular city of radius 5km. the density of the population at a distance r (in km) from this road is well approximated by d=20-4r (in thousands of people per square km). Find the total population of the city.

I'm told the answer is: 904,130............but, im not sure.

thanks


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mpdlc
guest
Mar-04-06, 06:59 PM (EST)
 
1. "RE: density question"
In response to message #0
 
   Just a quick answer, you do not need calculus to figured out.
Since density at the outer border of the city equal 0 and is a linear decreasing function of the radius. Your problem is like calculate the volume of a cone of r=5 and height=20. so the population must be

532,599 inhabitants


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mr_homm
Member since May-22-05
Mar-05-06, 11:12 AM (EST)
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2. "RE: density question"
In response to message #1
 
   >Just a quick answer, you do not need calculus to figured
>out.
>Since density at the outer border of the city equal 0 and is
>a linear decreasing function of the radius. Your problem is
>like calculate the volume of a cone of r=5 and height=20. so
>the population must be
>
>532,599 inhabitants

Unfortunately, this is not right. You are measuring distance from a point at the center of the circle. This would give the cone shape you are using. Instead, you have to measure distance from the diameter. If you use your idea of doing three dimensional geometry, the shape is more like the end of a chisel, not a cone.

I have tried to see a geometric way to get the volume, but I can't find one. I went ahead and did the integral, which was messy (you have to change variables twice). The answer I got was
2rrd(pi/4 - 1/3), where r is the radius and d is the maximum density. In this problem, r=5 and d=40. The answer I get is 904.1296 in units of thousands of people. Rounding off to the nearest whole person gives 904,130.

--Stuart Anderson


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mpdlc
guest
Mar-06-06, 08:15 PM (EST)
 
3. "RE: density question"
In response to message #2
 
   Hi Stuart,

I think you are right, I just was too quickie in reading the question.

Your answer is correct.

The body of the solid I envision, it has just the shape of a that kind of multipurpose plug that can in the puzzle books and it can be used to close three different kind of holes circular, isosceles triangle or a rectangular hole, located inside a crystal box the three main views are a circle, a rectangle and a triangle.

It has the very same shape of a cylinder if we remove two cylindrical wedges. So its volume will be the cylinder volume Vc = pi*Hm*R^2 , less the two wedges volume. Vw= 2*Hm*R^2 times two .

Which amount obviously to Vn= Hm*R^2(pi-4/3)

Making Hm=20, and R=5.

You get the nine hundred and four thousand... you posted. The only inacuracy in your formula is pi/4 should be pi/2.

For me using calculus, is the easier way to go, probably Archimedes got it without it, and maybe we can get it too there,.. it will take a lot of ingenuity.

Using calculus the answer is almost immediate, we can calculate either the half of our solid or even easier the cylindrical wedges.

To obtain the volume either the solid or the wedges, in both cases the trick is slicing the solid vertically and parallel to the given diameter we obtain rectangular sections for the cylindrical wedges, below it is the calculus for each wedge

Doing the slicing as indicated a general section at distance r of the diameter will be

S(r) = (Hm/R))*r*(R^2-r^2)^.5

We must integrate S(r) between 0 and R.

I know you can do it close eyed, either directly, using trig or even easier making the change w=(r/R)^2 , upper limit 1; lower limit 0...

The answer for each wedge is Vw=2/3 Hm*R^2

But I am still thinking in how it might be obtained without the calculus help.


Regards,

Mariano


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