Is it possible to fit more than 23 dots in a 7 by 8 rectangle?

Question

For practical reasons I would like to know how many dots you can fit in an $7 \times 8$ box with no two dots closer than 2 metres from each other.

The simplest arrangement has 4 dots along each row and 5 rows. Can you do any better?

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One method is to expand the box to $9 \times 10$ and fit circles of radius $1$. We can fit $23$ with:

But is it possible to fit more?

Answer 1

You can do at least 23, since $7/4=1.75 > \sqrt{3}$:

Answer 2

Yes, in this example fitting 23 circles of radius=1 into an 9x10 rectangle is optimal.

This hexagonal packing arrangement plus the close cropped edges yields the highest packing efficiency for circles into a rectangle.

Packing efficiency for 23 circles = 23xPi/(9x10) = 80.3%

Packing efficiency for 24 circles = 24xPi/(9x10) = 83.7%

Referring to this graph: https://i.stack.imgur.com/ziRBe.jpg it would appear that 83.7% is not possible. (Note: this graph allowed for a slight spacing between circles, I could recalculate the graph for zero spacing as per the original question but I think you get the idea)

For further details on similar problems: How many circles of a given radius can be packed into a given rectangular box?