Ex 6.5, 17 - A square piece of tin of side 18 cm is made into - Minima/ maxima (statement questions) - Geometry questions

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  1. Chapter 6 Class 12 Application of Derivatives
  2. Serial order wise

Transcript

Ex 6.5,17 A square piece of tin of side 18 cm is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible. Let š‘„ cm be the length of a side of the removed square Length after removing = 18 ā€“ š‘„ ā€“ š‘„ = 18 ā€“ 2š‘„ Breadth after removing = 18 ā€“ š‘„ ā€“ š‘„ = 18 ā€“ 2š‘„ Height of the box = š‘„ We need to maximize volume of box Let V be the volume of a box Ex 6.5,17 A square piece of tin of side 18 cm is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible. Let side cut off be š‘„ cm Length after cutting = 18 ā€“ š‘„ ā€“ š‘„ = 18 ā€“ 2š‘„ Breadth after cutting = 18 ā€“ š‘„ ā€“ š‘„ = 18 ā€“ 2š‘„ Height = š‘„ Volume = Length Ɨ Breadth Ɨ Height V =ļ·18āˆ’2š‘„ļ·Æ ļ·18āˆ’2š‘„ļ·Æ š‘„ Let V ļ·š‘„ļ·Æ = ļ·18āˆ’2š‘„ļ·Æ ļ·18āˆ’2š‘„ļ·Æ š‘„ We need to find Max volume as possible Hence we need to find Maximum value of V Volume of a cuboid = l Ɨ b Ɨ h Vļ·š‘„ļ·Æ=ļ·18āˆ’2š‘„ļ·Æļ·18āˆ’2š‘„ļ·Æļ·š‘„ļ·Æ Vļ·š‘„ļ·Æ=ļ·ļ·18āˆ’2š‘„ļ·Æļ·®2ļ·Æš‘„ Finding vā€™ļ·š‘„ļ·Æ Vā€™ ļ·š‘„ļ·Æ = ļ·š‘‘ļ·ļ·ļ·18āˆ’2š‘„ļ·Æļ·®2ļ·Æš‘„ļ·Æļ·®š‘‘š‘„ļ·Æ Vā€™ ļ·š‘„ļ·Æ = ļ·š‘‘ļ·ļ·ļ·18āˆ’2š‘„ļ·Æļ·®2ļ·Æļ·Æļ·®š‘‘š‘„ļ·Æ . š‘„ + ļ·š‘‘ļ·š‘„ļ·Æļ·®š‘‘š‘„ļ·Æ . ļ·ļ·18āˆ’2š‘„ļ·Æļ·®2ļ·Æ = ļ·2ļ·18āˆ’2š‘„ļ·Æ.ļ·š‘‘ļ·18āˆ’2š‘„ļ·Æļ·®š‘‘š‘„ļ·Æļ·Æš‘„+ļ·1.ļ·ļ·18āˆ’2š‘„ļ·Æļ·®2ļ·Æļ·Æ = ļ·2ļ·18āˆ’2š‘„ļ·Æ.ļ·0āˆ’2ļ·Æļ·Æš‘„+ ļ·ļ·18āˆ’2š‘„ļ·Æļ·®2ļ·Æ = ā€“4ļ·18āˆ’2š‘„ļ·Æš‘„+ļ·ļ·18āˆ’2š‘„ļ·Æļ·®2ļ·Æ = ļ·18āˆ’2š‘„ļ·Æ ļ·āˆ’4š‘„+ļ·18āˆ’2š‘„ļ·Æļ·Æ = ļ·18āˆ’2š‘„ļ·Æļ·18āˆ’6š‘„ļ·Æ Putting Vā€™ļ·š‘„ļ·Æ = 0 So, š‘„ = 3 , 9 If š‘„ = 9 Breadth of a box = 18 ā€“ 2š‘„ = 18 ā€“ 2(9) = 18 ā€“ 18 = 0 Since, breadth cannot be zero, ā‡’ x = 9 is not possible Hence š‘„ = 3 only Now finding Vā€™ā€™ ļ·š‘„ļ·Æ Vā€™ļ·š‘„ļ·Æ=ļ·18āˆ’2š‘„ļ·Æļ·18āˆ’6š‘„ļ·Æ Vā€™ ļ·š‘„ļ·Æ=ļ·š‘‘ļ·18āˆ’2š‘„ļ·Æļ·®š‘‘š‘„ļ·Æ.ļ·18āˆ’6š‘„ļ·Æ+ļ·š‘‘ļ·18āˆ’6š‘„ļ·Æļ·®š‘‘š‘„ļ·Æ.ļ·18āˆ’2š‘„ļ·Æ = ļ·0 āˆ’2ļ·Æļ·18āˆ’6š‘„ļ·Æ+ļ·āˆ’6ļ·Æļ·18āˆ’2š‘„ļ·Æ = ā€“2ļ·18āˆ’6š‘„ļ·Æāˆ’6ļ·18āˆ’2š‘„ļ·Æ = ā€“36 + 12x ā€“ 108 + 12x = 24x ā€“ 144 Putting š‘„ = 3 Vā€™ā€™(3) = 23 Ɨ 3 ā€“ 144 = ā€“ 72 < 0 Since Vā€™ā€™ļ·š‘„ļ·Æ<0 for š‘„=3 āˆ“ š‘„ = 3 is point of Maxima V(x) is Maximum at š‘„=3 Hence, 3 cm side of the square to be cut off so that the volume of the box is maximum

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.