Question 18 - End-of-Chapter Exercises - Chapter 6 Class 9 - Measuring Space: Perimeter and Area (Ganita Manjar

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Question 18 Use the above to make a conjecture about the area occupied by circles fitted into a rectangle in the manner shown. Test your conjecture for particular cases: 10 circles; 20 circles; 50 circles. Then prove your conjecture! Let’s consider n circles inside a rectangle Let Length of Rectangle = l Breadth of Rectangle = b Now, Diameter of circle = b ∴ Radius of circle = 𝒃/𝟐 And, n × Diameter = Length of rectangle n × b = l l = nb We need to find fraction of the rectangle is covered by the circles So, we need to find (𝑨𝒓𝒆𝒂 𝒐𝒇 𝒏 𝒄𝒊𝒓𝒄𝒍𝒆𝒔)/(𝑨𝒓𝒆𝒂 𝒐𝒇 𝑹𝒆𝒄𝒕𝒂𝒏𝒈𝒍𝒆) Now, (𝑨𝒓𝒆𝒂 𝒐𝒇 𝒏 𝒄𝒊𝒓𝒄𝒍𝒆𝒔)/(𝑨𝒓𝒆𝒂 𝒐𝒇 𝑹𝒆𝒄𝒕𝒂𝒏𝒈𝒍𝒆)=(𝑛 × 𝜋 × 𝑅𝑎𝑑𝑖𝑢𝑠^2)/(𝐿𝑒𝑛𝑔𝑡ℎ × 𝐵𝑟𝑒𝑎𝑑𝑡ℎ) =(𝑛 × 𝜋 × (𝑏/2)^2)/(𝑙 × 𝑏) =(𝒏 × 𝝅 × (𝒃/𝟐)^𝟐)/(𝒏𝒃 × 𝒃) =(𝜋 × (𝑏/2)^2)/𝑏^2 =(𝜋 ×𝑏^2/4)/𝑏^2 =(𝜋 × 𝑏^2)/(𝑏^2 × 4) =𝝅/𝟒 Thus, Fraction covered is 𝝅/𝟒 Let’s test for 10 circles; 20 circles; 50 circles TESTING PARTICULAR CASES CASE: 10 Circles CASE: 20 Circles CASE: 50 Circles 00000000 88888888888888 000000000000000000 (10πr^2)/(4(10)r^2 )=π/4 (20πr^2)/(4(20)r^2 )=π/4 (50πr^2)/(4(50)r^2 )=π/4 Result: Holds True. Result: Holds True. Result: Holds True.