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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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

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.