Question

A rectangular field is 40 meters long and 30 meters wide. Draw diagonals on this field and then draw circles of radius 1.25 meters, with centers only on the diagonals. Each circle must fall completely within the field. Any two circles can touch each other but should not overlap. What is the maximum number of such circles that can be drawn in the field?

Accepted Answer

D. Diagonal length = $\sqrt{40^2 + 30^2} = 50$ m. For a circle of radius $r = 1.25$ m touching two adjacent sides of the corner (angle $	heta$ at corner where $	an 	heta = 30/40 = 3/4$), $\sin(	heta/2)$ determines distance from vertex to center = $1.25 	imes \sqrt{1 + (3/4)^2}/... = 1.25 \sqrt{2}$ roughly. Usable diagonal length is $50 - 2 	imes 0.51 = 48.98$ m. Max circles per diagonal = $\lfloor 48.98 / 2.5 floor = 19$. Since 1 circle can be placed exactly at the intersection of the two diagonals (common to both), each half diagonal gets 9 circles. Total circles = $4 	imes 9 + 1 = 37$.

Maximum Circles on Rectangular Field Diagonals

Explanation

Practise this under exam conditions