Two circular pieces of equal radii and maximum area, touching each other are cut out from a rectangular cardboard of dimensions 14 cm × 7 cm. Find the area of the remaining cardboard.

Given:

Length of the rectangular cardboard = 14 cm

Breadth of the rectangular cardboard = 7 cm

∵ Area of rectangle = length × breadth

∴ Area of cardboard = 14 × 7 = 98 cm^{2}

Let the two circles with equal radii and maximum area have a radius of ‘r’ cm each.

Then, 2r = 7

r = cm

∵ Area of circle = πr^{2}

∴ Area of two circular cut outs = 2 × πr^{2}

= 2 × × ×

= 11 × 7 = 77 cm^{2}

Thus, the area of remaining cardboard = 98 – 77 = 21 cm^{2}

__Hence, the area of remaining cardboard is 21 cm ^{2}.__

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