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Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


Ex 6.3, 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 š‘„ be the length of a side of the removed square Thus, Length after removing = 45 ā€“ š‘„ ā€“š‘„ = 45 ā€“ 2š‘„ Breadth after removing = 24 ā€“š‘„ ā€“š‘„ = 24 ā€“ 2š‘„ Height of the box = š‘„ We need to maximize volume of box Let V be the volume of a box Volume of a cuboid = l Ɨ b Ɨ h = (45āˆ’2š‘„)(24āˆ’2š‘„)(š‘„) = [45(24āˆ’2š‘„)āˆ’2š‘„(24āˆ’2š‘„)]š‘„ = (1080āˆ’90š‘„āˆ’48š‘„+4š‘„^2 )š‘„ = 1080š‘„āˆ’90š‘„^2āˆ’48š‘„^2+4š‘„^3 = 1080š‘„āˆ’138š‘„^2+4š‘„^3 = 2(540š‘„āˆ’69š‘„^2+2š‘„^3 ) = 2(2š‘„^3āˆ’69š‘„^2+540š‘„) V = 2(2š‘„^3āˆ’69š‘„^2+540) Diff w.r.t š‘„ Vā€™(š‘„)=2š‘‘[2š‘„^3āˆ’ 69š‘„^2+ 540š‘„]/š‘‘š‘„ = 2[6š‘„^2āˆ’69 Ɨ2š‘„+540] = 2[6š‘„^2āˆ’138š‘„+540] = 2 Ɨ 6[š‘„^2āˆ’23š‘„+90] = 12[š‘„^2āˆ’23š‘„+90] Putting Vā€™(š‘„)=0 12(š‘„^2āˆ’23š‘„+90)=0 š‘„^2āˆ’23š‘„+90=0 š‘„^2āˆ’5š‘„āˆ’18š‘„+90=0 š‘„(š‘„āˆ’5)āˆ’18(š‘„āˆ’5)=0 (š‘„āˆ’18)(š‘„āˆ’5)=0 So, x = 18 & x = 5 If š‘„ = 18 Breadth of a box = 24 ā€“ 2š‘„ = 24 ā€“ 2(18) = 24 ā€“ 36 = ā€“12 Since, breadth cannot be negative, ā‡’ x = 18 is not possible Hence š‘„ = 5 only Finding Vā€™ā€™(š‘„) Vā€™(š‘„)=12(š‘„^2āˆ’23š‘„+90) Vā€™ā€™(š‘„)=12(2š‘„āˆ’23) Putting š‘„=5 Vā€™ā€™(5)=12(2(5)āˆ’23)= 12(10āˆ’23)= 12(āˆ’ 13)= ā€“ 156 Vā€™ā€™(š‘„)<0 when š‘„=5 Thus, V(š‘„) is maximum at š‘„=5 āˆ“ Square of side 5 cm is cut off from each Corner

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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 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.