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Last updated at Jan. 7, 2020 by Teachoo

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Misc 3 The two equal sides of an isosceles triangle with fixed base b are decreasing at the rate of 3 cm per second. How fast is the area decreasing when the two equal sides are equal to the base ? Let x be the equal sides isosceles triangle with fixed base b. i.e. AB = AC = ๐ฅ & BC = b Given that side of triangle decreasing when ๐ฅ = b i.e. ๐๐ฅ/๐๐ก= 3 cm/sec. We need to find how fast area is decreasing when ๐ฅ = b i.e. ๐๐ด/๐๐ก when ๐ฅ = b Finding Area Draw a perpendicular AD to BC โ i.e. AD โฅ BC In isosceles triangle, perpendicular from vertex to the side bisects the side i.e. D is the mid point of BC BD = DC โด BD = DC = ๐/2 In โ ๐ด๐ท๐ต Using Pythagoras theorem (๐ด๐ต)^2=(๐ด๐ท)^2+(๐ต๐ท)^2 (๐ฅ)^2=(๐ด๐ท)^2+ (๐/2)^2 ๐ฅ2 โ (๐/2)^2=(๐ด๐ท)^2 (๐ด๐ท)^2 = ๐ฅ2 โ (๐/2)^2 ๐ด๐ท=โ(๐ฅ2โ(๐/2)^2 ) We know that Area of isosceles triangle = 1/2 ร Base ร Height A = 1/2 ร b ร โ(๐ฅ2โ(๐/2)^2 ) A = 1/2 ร b ร /2 โ(๐ฅ2โ๐^2/4) We need ๐๐ด/๐๐ก Diff w.r.t.x ๐๐ด/๐๐ก= 1/2 ๐ . ๐(โ(๐ฅ^2 โ ๐^2/4))/๐๐ก ๐๐ด/๐๐ก= 1/2 ๐ [1/(2โ(๐ฅ2 โ ๐^2/4)) ร ๐(๐ฅ^2 โ ๐^2/4)/๐๐ก] ๐๐ด/๐๐ก= 1/2 ๐ [1/(2โ(๐ฅ2 โ ๐^2/4)) ร(๐(๐ฅ2)/๐๐กโ0)] ๐๐ด/๐๐ก= 1/2 ๐ [1/(2โ(๐ฅ2 โ ๐^2/4)) ร(๐(๐ฅ2)/๐๐ฅ ร ๐๐ฅ/๐๐ก)] ๐๐ด/๐๐ก= 1/2 ๐ [1/(2โ(๐ฅ2 โ ๐^2/4)) ร 2๐ฅ ร ๐๐ฅ/๐๐ก] ๐๐ด/๐๐ก= 1/2 ๐ [1/(2โ(๐ฅ2 โ ๐^2/4)) ร 2๐ฅ ร 3] ๐๐ด/๐๐ก= ๐/(4โ(๐ฅ2 โ ๐^2/4)) ร 6๐ฅ Finding ๐๐ด/๐๐ก At ๐ฅ = b โ ๐๐ด/๐๐กโค|_(๐ฅ = ๐)=(6๐^2)/(4โ(๐^2 โ ๐^2/4))= (6๐^2)/(4โ((4๐^2 โ ๐^2)/4))= (6๐^2)/(4โ((3๐^2)/4))= (6๐^2)/((4โ3 ๐)/2)= (6๐^2)/(2โ3 ๐) = 3๐/โ3=๐โ3 Since dimension of area is cm2 s โ ๐๐ด/๐๐ก at ๐ฅ = b is ๐โ๐ cm2/s

Miscellaneous

Misc 1
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Not in Syllabus - CBSE Exams 2021

Misc 2 Important

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Misc 4

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Misc 6 Important

Misc 7

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Misc 10

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Misc 12 Important

Misc 13 Important

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Misc 15 Important

Misc 16 Important

Misc 17 Important

Misc 18 Important

Misc. 19 Not in Syllabus - CBSE Exams 2021

Misc 20 Important

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Misc 22

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Misc 24 Important

Chapter 6 Class 12 Application of Derivatives

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About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.