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Question

On a square cardboard sheet of area 784cm2, four congruent circular plates of maximum size are placed such that each circular plate touches the other two plates and each side of the square sheet is tangent to two circular plates. Find the area of the square sheet not covered by the circular plates.


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Solution

According to the question,

Step 1: Calculate the length of the side of the square cardboard:

Area of the square cardboard = 784cm2

Area of square = side2

side2 = 784cm2

side=784side=28cm

Step 2: Calculate the radius of the circle :

Since circles are touching each other and the sides of the square is tangent to the circle,

therefore, sideofthesquare=2×diameterofthecircle

2×diameterofthecircle=28diameterofthecircle=28÷2=14

Radius of the circle = Diameter2

= 142=7cm

Step 3: Calculate the area of shaded portion:

Area of four circles = 4×πr2

=4×227×72=4×227×7×7=88×7=616cm2

Area of shaded portion =Areaofcardboard-Areaoffourcircles

=784-616=168cm2

Hence, the area of shaded portion is 168cm2.


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