# Math Help - practical problem

1. ## practical problem

Hi there,

Can anyone solve, or help me to solve a practical problem that I have.
I have a rectangle which is 400mm by 300mm, I need to pack 40 equal circles of diameter 45mm onto this rectangle. I need to maximise the minimum distance between the circles. Another way of putting this is to say that I want the 40 circles to be as far away from each other as possible but still fit into my rectangle. Can anyone help with this?

Regards

Frankie

2. As your rectangle is fixed you should find the radius of your circles to detemine how many will fit. The sum of radii both longways and up and down will give you a length and width

$r= \frac{45}{2\pi}$

3. Place 8 along the 400 length: do this 5 times; that'll be total of 40 circles;
so you have 5 rows of 8 circles.
Stick the "outermost" 22 circles tangent to the rectangle's perimeter, with
the 4 in the corners tabgent to 2 of the sides.

8 * 45 = 360 ; 400 - 360 = 40 ; leave 40/7 as horizontal length in between circles.

5 * 45 = 225 ; 300 - 225 = 75 ; leave 75/4 as vertical length in between circles.