




Miscellaneous
Misc 1 (b) Important Deleted for CBSE Board 2022 Exams
Misc 2 Important
Misc 3 Important
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
Misc 17 Important You are here
Misc 18 Important
Misc. 19 (MCQ) Deleted for CBSE Board 2022 Exams
Misc 20 (MCQ) Important
Misc 21 (MCQ) Important
Misc 22 (MCQ)
Misc. 23 (MCQ) Important
Misc 24 (MCQ) Important
Last updated at April 19, 2021 by Teachoo
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.Given Radius of sphere = R Let h be the height & π be the diameter of cylinder In β π¨π©πͺ 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 π2 = 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)^2Γ β V = Ο Γ π₯^2/4Γ β V = Ο ((4π ^2 β β^2 ))/4 Γ β V = (4π ^2 πβ)/4β(πβ^3)/4 V = ΟhR2 β (π π^π)/π Differentiating w.r.t π ππ/πβ=π(πβπ ^2 β πβ^3/4)/πβ ππ/πβ= ΟR2 π(β)/πββπ/4 π(β^3 )/πβ ππ΅/πβ= ΟR2 β π/4 (3β^2 ) ππ/πβ= ΟR2 β 3π/4 h2 Putting π π½/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 = ππΉ/βπ Finding (π ^π π½)/(π π^π ) ππ/πβ=ππ ^2β3/(4 ) π β^2 Differentiating w.r.t. h (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 ^π π½)/(π π^π )=(βππ π)/π Since (π ^π π½)/(π π^π )<π for h = 2π /β3 β΄ Volume is maximum for h = 2π /β3 We also need to find Maximum Volume V = ΟhR2 β (πβ^3)/4 V = ΟR2 Γ 2π /β3 β π/4 Γ (2π /β3)^3 V = (2ππ ^3)/β3 β π/4 Γ(8π ^3)/(3β3) V = (2ππ ^3)/β3 β (2ππ ^3)/(3β3) V = (2ππ ^3)/β3 (1β1/3) V = (2ππ ^3)/β3 Γ2/3 V = (ππ πΉ^π)/(πβπ) cubic unit