Check Full Chapter Explained - Continuity and Differentiability - Application of Derivatives (AOD) Class 12

Last updated at Jan. 7, 2020 by Teachoo

Check Full Chapter Explained - Continuity and Differentiability - Application of Derivatives (AOD) Class 12

Transcript

Misc 17 Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is 2π /β3 . Also find the maximum volume. Let R be the radius of sphere Let h be the height & π₯ be the diameter of cylinder In β π΄π΅πΆ Using Pythagoras theorem (πΆπ΅)^2+(π΄π΅)^2=(π΄πΆ)^2 h2 + (π₯)^2=(π +π )^2 h2 + π₯2 =(2π )^2 h2 + π₯2 = 4R2 π₯2 = 4R2 β h2 We need to find maximum volume of cylinder Let V be the volume of cylinder V = Ο (πππππ’π )^2Γ(βπππβπ‘) V = Ο (π₯/2)^2Γβ V = Ο Γ π₯^2/4Γβ V = Ο ((4π ^2 β β^2 ))/4 Γβ β¦(1) V = Ο ((4π ^2 β β^2 ))/4 Γβ V = (4π ^2 πβ)/4β(πβ^3)/4 V = ΟhR2 β (πβ^3)/4 Different w.r.t β ππ£/πβ=π(πβπ ^2 β πβ^3/4)/πβ ππ£/πβ= Ο R2 π(β)/πββπ/4 π(β^3 )/πβ ππ£/πβ= Ο R2 β π/4 (3β^2 ) ππ£/πβ= Ο R2 β 3π/4 h2 Putting π π/π π=π Ο R2 β 3/4 π β^2=0 3/4 πβ^2=ππ ^2 h2 = (ππ ^2)/(3/4 π) h2 = (4π ^2)/3 h =β((4π ^2)/3) h = 2π /β3 Finding (π ^π π)/(π π^π ) ππ£/πβ=ππ ^2β3/(4 ) π β^2 Differentiating w.r.t.x (π^2 π£)/(πβ^2 )= π(ππ ^2 β 3/4 πβ^2 )/πβ (π^2 π£)/(πβ^2 )= 0 β 3π/4 Γ2β (π^2 π£)/(πβ^2 )= (β3πβ)/2 Putting h = 2π /β3 (π^2 π£)/(πβ^2 ) = (β3π)/2 (2π /β3) = ββ3ππ < 0 β΄ h = 2π /β3 is point of maxima So, volume is maximum when h = 2π /β3 From (1) π₯2 = 4R2 β h2 π₯2 = 4R2 β (2π /β3)^2 π₯2 = 4R2 β (4π ^2)/3 π₯2 = (12π ^2 β 4π ^2)/3 π₯2 = 8/(3 ) γ π γ^2 Maximum value of volume is V = Ο (π₯/2)^2 β V = π/4 π₯^2 β Putting value of π₯2 & h V = π/4 Γ (8π ^2)/3 Γ 2π /β3 V = ( 16ππ ^3)/(12β3) V = (ππ πΉ^π)/(πβπ) cubic unit

Miscellaneous

Misc 1
Important
Not in Syllabus - CBSE Exams 2021

Misc 2 Important

Misc 3 Important Not in Syllabus - CBSE Exams 2021

Misc 4

Misc 5 Important

Misc 6 Important

Misc 7

Misc 8 Important

Misc 9 Important

Misc 10

Misc 11 Important

Misc 12 Important

Misc 13 Important

Misc 14 Important

Misc 15 Important

Misc 16 Important

Misc 17 Important You are here

Misc 18 Important

Misc. 19 Not in Syllabus - CBSE Exams 2021

Misc 20 Important

Misc 21 Important

Misc 22

Misc. 23 Important

Misc 24 Important

Chapter 6 Class 12 Application of Derivatives

Serial order wise

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.