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Question 37 (iii) (Choice 2) - CBSE Class 12 Sample Paper for 2023 Boards - Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

Last updated at April 16, 2024 by

Use Second Derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area.

 

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Transcript

Question 37 (iii) (Choice 2) Use Second Derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area.Finding (𝒅^𝟐 𝒁)/(𝒅𝒙^𝟐 ) 𝑑𝑍/𝑑𝑥=(32𝑏^2)/𝑎^2 × (𝑎^2 𝑥−2𝑥^3) Differentiating w.r.t 𝑥 (𝑑^2 𝑍)/(𝑑𝑥^2 )=(32𝑏^2)/𝑎^2 × (𝑎^2−2 × 3𝑥^2) (𝒅^𝟐 𝒁)/(𝒅𝒙^𝟐 )=(𝟑𝟐𝒃^𝟐)/𝒂^𝟐 × (𝒂^𝟐−𝟔𝒙^𝟐) Putting x = 𝒂/√𝟐 (𝑑^2 𝑍)/(𝑑𝑥^2 )=(32𝑏^2)/𝑎^2 × (𝑎^2−6(𝑎/√2)^2 ) (𝑑^2 𝑍)/(𝑑𝑥^2 )=(32𝑏^2)/𝑎^2 × (𝑎^2−6 ×𝑎^2/2) (𝑑^2 𝑍)/(𝑑𝑥^2 )=(32𝑏^2)/𝑎^2 × (𝑎^2−3𝑎^2 ) (𝑑^2 𝑍)/(𝑑𝑥^2 )=(32𝑏^2)/𝑎^2 × −2𝑎^2 < 0 Since 𝐙^′′ < 0 for x = 𝒂/√𝟐 ∴ Z is maximum when x = 𝒂/√𝟐 Thus, A is maximum at x = 𝒂/√𝟐 Finding length 2x and 2y Length = 2x = 2 × 𝑎/√2 = √𝟐a Breadth = 2y = 2 × (" " 𝒃)/𝒂 √((𝒂^𝟐 − 𝒙^𝟐 ) ) = 2b/𝑎 ×√(𝑎^2−(𝒂/√𝟐)^2 ) = 2b/𝑎 ×√(𝑎^2−𝑎^2/2) = 2b/𝑎 ×√(𝑎^2/2) = 2b/𝑎 ×𝒂/√𝟐 = 2b/√𝟐 = √𝟐 𝒃

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.