# Ex 6.5,18

Last updated at March 11, 2017 by Teachoo

Last updated at March 11, 2017 by Teachoo

Transcript

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 𝑥 m be the length of a side of the removed square Length after removing = 45 – 𝑥 – 𝑥 = 45 – 2𝑥 Breadth after removing = 24 – 𝑥 – 𝑥 = 24 – 2𝑥 Height of the box = 𝑥 We need to maximize volume of box Let V be the volume of a box Volume of a cuboid = l × b × h = 45−2𝑥 24−2𝑥 𝑥 = 45 24−2𝑥−2𝑥 24−2𝑥𝑥 = 1080−90𝑥−48𝑥+4 𝑥2𝑥 = 1080𝑥−90 𝑥2−48 𝑥2+4 𝑥3 = 1080𝑥−138 𝑥2+4 𝑥3 = 2 540𝑥−69 𝑥2+2 𝑥3 = 2 2 𝑥3−69 𝑥2+540𝑥 V = 2 2 𝑥3−69 𝑥2+540 Diff w.r.t 𝑥 V’ 𝑥= 2𝑑 2 𝑥3− 69 𝑥2+ 540𝑥𝑑𝑥 = 2 6 𝑥2−69 ×2𝑥+540 = 2 6 𝑥2−138𝑥+540 = 2 × 6 𝑥2−23𝑥+90 = 12 𝑥2−23𝑥+90 Putting V’ 𝑥=0 12 𝑥2−23𝑥+90=0 𝑥2−23𝑥+90=0 𝑥2−5𝑥−18𝑥+90=0 𝑥 𝑥−5−18 𝑥−5=0 𝑥−18 𝑥−5=0 So, x = 18 & x = 5 If 𝑥 = 18 Breadth of a box = 24 – 2𝑥 = 24 – 2(18) = 24 – 36 = –12 Since, breadth cannot be negative, ⇒ x = 18 is not possible Hence 𝑥 = 5 only Finding V’’ 𝑥 V’ 𝑥=12 𝑥2−23𝑥+90 V’’ 𝑥=12 2𝑥−23 Putting 𝑥=5 V’’ 5=12 2 5−23= 12 10−23= 12 − 13= – 156 V’’ 𝑥<0 when 𝑥=5 Thus, V 𝑥 is maximum at 𝑥=5 ∴ Square of side 5 cm is cut off from each Corner

Ex 6.3,5
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Ex 6.3,7 Important

Ex 6.3,12 Important

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Example 40 Important

Ex 6.5,1 Important

Ex 6.5,5 Important

Ex 6.5,7 Important

Ex 6.5,11 Important

Ex 6.5,18 Important You are here

Ex 6.5,20 Important

Ex 6.5,23 Important

Ex 6.5,26 Important

Ex 6.5,28 Important

Example 46 Important

Example 47 Important

Misc 6 Important

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Class 12

Important Question for exams Class 12

- Chapter 1 Class 12 Relation and Functions
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- Chapter 6 Class 12 Application of Derivatives
- Chapter 7 Class 12 Integrals
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- Chapter 12 Class 12 Linear Programming
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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 7 years. He provides courses for Mathematics and Science from Class 6 to 12. You can learn personally from here https://www.teachoo.com/premium/maths-and-science-classes/.