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Last updated at Jan. 7, 2020 by Teachoo

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Ex 6.5, 18 A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum ? Let ๐ฅ be the length of a side of the removed square Thus, Length after removing = 45 โ ๐ฅ โ๐ฅ = 45 โ 2๐ฅ Breadth after removing = 24 โ๐ฅ โ๐ฅ = 24 โ 2๐ฅ Height of the box = ๐ฅ We need to maximize volume of box Let V be the volume of a box Volume of a cuboid = l ร b ร h = (45โ2๐ฅ)(24โ2๐ฅ)(๐ฅ) = [45(24โ2๐ฅ)โ2๐ฅ(24โ2๐ฅ)]๐ฅ = (1080โ90๐ฅโ48๐ฅ+4๐ฅ^2 )๐ฅ = 1080๐ฅโ90๐ฅ^2โ48๐ฅ^2+4๐ฅ^3 = 1080๐ฅโ138๐ฅ^2+4๐ฅ^3 = 2(540๐ฅโ69๐ฅ^2+2๐ฅ^3 ) = 2(2๐ฅ^3โ69๐ฅ^2+540๐ฅ) V = 2(2๐ฅ^3โ69๐ฅ^2+540) Diff w.r.t ๐ฅ Vโ(๐ฅ)=2๐[2๐ฅ^3โ 69๐ฅ^2+ 540๐ฅ]/๐๐ฅ = 2[6๐ฅ^2โ69 ร2๐ฅ+540] = 2[6๐ฅ^2โ138๐ฅ+540] = 2 ร 6[๐ฅ^2โ23๐ฅ+90] = 12[๐ฅ^2โ23๐ฅ+90] Putting Vโ(๐ฅ)=0 12(๐ฅ^2โ23๐ฅ+90)=0 ๐ฅ^2โ23๐ฅ+90=0 ๐ฅ^2โ5๐ฅโ18๐ฅ+90=0 ๐ฅ(๐ฅโ5)โ18(๐ฅโ5)=0 (๐ฅโ18)(๐ฅโ5)=0 So, x = 18 & x = 5 Thus, V(๐ฅ) is maximum at ๐ฅ=5 โด Square of side 5 cm is cut off from each Corner

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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.