Ex 6.5, 18 - A rectangular sheet of tin 45 cm by 24 cm is made

Advertisement

Ex 6.5,18 - Chapter 6 Class 12 Application of Derivatives - Part 2

Advertisement

Ex 6.5,18 - Chapter 6 Class 12 Application of Derivatives - Part 3 Ex 6.5,18 - Chapter 6 Class 12 Application of Derivatives - Part 4 Ex 6.5,18 - Chapter 6 Class 12 Application of Derivatives - Part 5

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

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 ๐‘ฅ be the length of a side of the removed square Thus, Length after removing = 45 โ€“ ๐‘ฅ โ€“๐‘ฅ = 45 โ€“ 2๐‘ฅ Breadth after removing = 24 โ€“๐‘ฅ โ€“๐‘ฅ = 24 โ€“ 2๐‘ฅ Height of the box = ๐‘ฅ We need to maximize volume of box Let V be the volume of a box Volume of a cuboid = l ร— b ร— h = (45โˆ’2๐‘ฅ)(24โˆ’2๐‘ฅ)(๐‘ฅ) = [45(24โˆ’2๐‘ฅ)โˆ’2๐‘ฅ(24โˆ’2๐‘ฅ)]๐‘ฅ = (1080โˆ’90๐‘ฅโˆ’48๐‘ฅ+4๐‘ฅ^2 )๐‘ฅ = 1080๐‘ฅโˆ’90๐‘ฅ^2โˆ’48๐‘ฅ^2+4๐‘ฅ^3 = 1080๐‘ฅโˆ’138๐‘ฅ^2+4๐‘ฅ^3 = 2(540๐‘ฅโˆ’69๐‘ฅ^2+2๐‘ฅ^3 ) = 2(2๐‘ฅ^3โˆ’69๐‘ฅ^2+540๐‘ฅ) V = 2(2๐‘ฅ^3โˆ’69๐‘ฅ^2+540) Diff w.r.t ๐‘ฅ Vโ€™(๐‘ฅ)=2๐‘‘[2๐‘ฅ^3โˆ’ 69๐‘ฅ^2+ 540๐‘ฅ]/๐‘‘๐‘ฅ = 2[6๐‘ฅ^2โˆ’69 ร—2๐‘ฅ+540] = 2[6๐‘ฅ^2โˆ’138๐‘ฅ+540] = 2 ร— 6[๐‘ฅ^2โˆ’23๐‘ฅ+90] = 12[๐‘ฅ^2โˆ’23๐‘ฅ+90] Putting Vโ€™(๐‘ฅ)=0 12(๐‘ฅ^2โˆ’23๐‘ฅ+90)=0 ๐‘ฅ^2โˆ’23๐‘ฅ+90=0 ๐‘ฅ^2โˆ’5๐‘ฅโˆ’18๐‘ฅ+90=0 ๐‘ฅ(๐‘ฅโˆ’5)โˆ’18(๐‘ฅโˆ’5)=0 (๐‘ฅโˆ’18)(๐‘ฅโˆ’5)=0 So, x = 18 & x = 5 If ๐‘ฅ = 18 Breadth of a box = 24 โ€“ 2๐‘ฅ = 24 โ€“ 2(18) = 24 โ€“ 36 = โ€“12 Since, breadth cannot be negative, โ‡’ x = 18 is not possible Hence ๐‘ฅ = 5 only Finding Vโ€™โ€™(๐‘ฅ) Vโ€™(๐‘ฅ)=12(๐‘ฅ^2โˆ’23๐‘ฅ+90) Vโ€™โ€™(๐‘ฅ)=12(2๐‘ฅโˆ’23) Putting ๐‘ฅ=5 Vโ€™โ€™(5)=12(2(5)โˆ’23)= 12(10โˆ’23)= 12(โˆ’ 13)= โ€“ 156 Vโ€™โ€™(๐‘ฅ)<0 when ๐‘ฅ=5 Thus, V(๐‘ฅ) is maximum at ๐‘ฅ=5 โˆด Square of side 5 cm is cut off from each Corner

About the Author

Davneet Singh's photo - Teacher, Engineer, Marketer
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.