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?

• A. 38
• B. 39
• C. 36
• D. 37
• E. 40

Answer 1

The Correct Option is D

A field is 40 meters long and 30 meters wide. Calculate the length of the diagonal using the Pythagorean theorem: a² + b² = c², where a and b are the sides of the rectangle, and c is the diagonal.
a = 40, b = 30.
Thus, c = √(40² + 30²) = √(1600 + 900) = 50 meters.

We need circles with centers on the diagonals and radius 1.25m. The diameter of each circle is 2 × 1.25 = 2.5 meters.

For circles not to overlap, the centers must be at least a diameter (2.5m) apart on the diagonal which is 50 meters long.
The number of circle centers that can fit = floor(50 / 2.5) = 20 per diagonal.

Since the rectangle has two diagonals that intersect, consider the intersection point where both diagonals can share a common circle. Therefore, total circles = (20 + 20) - 1 = 39.

However, to optimize circle placement, ensure each circle remains fully within field boundaries. To ensure separation at ends (consider half-circle allowance at each end):
Available length for center placement on each diagonal = 50 - 2 × 1.25 = 47.5 meters.
Max circles with adjusted placement: floor(47.5 / 2.5) = 19 per diagonal.
Thus, for combined diagonals: 19 + 19 circles - 1 shared = 37.

The maximum number of circles is: 37.