As you can see, this is an overestimate, because we aren't using the space around the edges of the packing as efficiently as possible. If all circles have area $10$, then at most $3659$ circles can fit in that area. If the rectangle is $257 \times 157$ and the radius of a circle is $\sqrt \approx 36592.5$. (Also, if the rectangle is only $2m \cdot r$ units tall, we can alternate columns with $m$ and $m-1$ circles.) So if you want the triangular packing to have $m$ circles in each column, and $n$ columns, then the rectangle must be at least $(2m 1) \cdot r$ units tall and $(2 (n-1)\sqrt3) \cdot r$ units long. Each pair of vertical blue lines is a distance $r \sqrt 3$ apart, and they're still a distance $r$ from the edges. If the circles have radius $r$, then each pair of horizontal red lines is a distance $r$ apart, and they're a distance $r$ from the edges. Giving the profit of each circle is: P(a) = 200 - 200/a (a is the area of the circle)Ĭonsider the following diagram of a triangular packing: So my question is: Did I calculate it in a correct way? Are there any other more effective calculation methods?īecause in later question, it asks me to find the area of the circle to so that we get the maximum profit. However, I find my math calculation kinda inefficient, long, and not correct in any other cases. Therefore the proportion of the plane covered by the circles is pi/4 0.785398ldots 78.5 to 3 significant figures. > That means in this case, i can fit in 43*72= 3096 circlesĢ) Then I try triangular pattern, which can fit more circles, 3575 circles. I had the height 157/d (diameter) -> I got about 43.999 -> So along the height, i can place 43 circle.I had the width 257/d (diameter) -> I got about 72.024 -> So along the width, i can place 72 circle. Packing problems for regular shapes (circles and rectangles) of objects and/or containers are well studied (see, e.g., a review 12 for circle packing).So, i try to pack as many as possible (taking this website as reference):ġ) First, I tried to place them in rectangular pattern: After a lot of research, I found out that there are no optimal solution. I'm asked to pack the maximum number of 10m^2 circle into a 257 x 157m rectangle.
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