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?
Explanation:
Length of the diagonal = √(302+402 ) = 50 m.
Each circle on the end of the diagonal will touch sides of the rectangular field
Using Pythagoras' theorem, the distance between the vertex of the rectangle and centre of the first circle drawn on the diagonal (OC) = 1.25√2
Distance between the vertex of the rectangle and circumference of the first circle drawn on the diagonal (OD) = 1.25√2 - 1.25 = 0.51 meters
Space that cannot be used to draw circle otherwise they will go outside rectangle on every diagonal = 0.51 × 2 = 1.02 meters
Space that can be used to draw circles = length of diagonal - unused space = 50 - 1.02 = 48.98 meters
On every diagonal, maximum number of such circles = usable length/diameter of each circle = 48.98/2.5 = 19.6
∴ Maximum 19 circles can be drawn on a diagonal.
Now, on every diagonal, one circle will be at the centre (intersection of diagonals) and 9 circles will be on each half of the diagonal
⇒ The circle in centre will be common for both diagonals.
∴ Total circles = 19 + 19 – 1 = 37.
Hence, option (c).
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