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Finding rate of change

Ex 6.1, 14 - Sand is pouring from a pipe at rate of 12 cm3/s

Ex 6.1,14 - Chapter 6 Class 12 Application of Derivatives - Part 2
Ex 6.1,14 - Chapter 6 Class 12 Application of Derivatives - Part 3
Ex 6.1,14 - Chapter 6 Class 12 Application of Derivatives - Part 4
Ex 6.1,14 - Chapter 6 Class 12 Application of Derivatives - Part 5


Transcript

Ex 6.1, 14 Sand is pouring from a pipe at the rate of 12 cm3/s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm? Given that sand is pouring from a pipe & falling sand forms a cone Let 𝒓 be the radius & 𝒉 be height of the sand cone & V be the volume of cone Also, Sand is pouring from a pipe at the rate of 12π‘π‘š^3/sec i.e. Rate of volume of a cone w.r.t time is 12π‘π‘š^3/sec i.e. 𝒅𝑽/𝒅𝒕 ="12" π’„π’Ž^πŸ‘ "/sec" We need to find how fast height of the Cone is increasing when height is 4cm i.e. find 𝒅𝒉/𝒅𝒕 when 𝒉=πŸ’π’„π’Ž Given that sand forms cone on the ground in such a way that the height of the cone is always one sixth of the radius i.e. β„Ž=1/6 π‘Ÿ 6β„Ž=π‘Ÿ 𝒓=πŸ”π’‰ We know that Volume of a cone = 1/3 πœ‹(π‘Ÿ^2 )β„Ž V = 1/3 Ο€γ€– 𝒓〗^2 β„Ž V = 1/3 Ο€ (πŸ”π’‰)^2 β„Ž V = 1/3 Ο€ Γ— 36β„Ž^2 Γ—β„Ž V = 1/3 Ο€ Γ— 36 β„Ž^3 V = 12Ο€ 𝒉^πŸ‘ (β–ˆ("From (2)" : π‘Ÿ" = 6" β„Ž)) Differentiating w.r.t 𝑑 𝑑𝑉/𝑑𝑑=𝑑(12πœ‹β„Ž^3 )/𝑑𝑑 𝑑𝑉/𝑑𝑑=12πœ‹ 𝑑(β„Ž^3 )/𝑑𝑑 𝑑𝑉/𝑑𝑑=12πœ‹ ×𝑑(β„Ž^3 )/𝑑𝑑 Γ— π‘‘β„Ž/π‘‘β„Ž 𝑑𝑉/𝑑𝑑=12πœ‹ ×𝒅(𝒉^πŸ‘ )/𝒅𝒉 Γ— π‘‘β„Ž/𝑑𝑑 𝒅𝑽/𝒅𝒕=12πœ‹ Γ— 3β„Ž^2 Γ— π‘‘β„Ž/𝑑𝑑 𝟏𝟐=12πœ‹ Γ— 3β„Ž^2 Γ— π‘‘β„Ž/𝑑𝑑 12/(12πœ‹ Γ— 3β„Ž^2 )=π‘‘β„Ž/𝑑𝑑 𝒅𝒉/𝒅𝒕=𝟏/(πŸ‘π…π’‰^𝟐 ) (From (1): 𝒅𝑽/𝒅𝒕 ="12" ) Putting β„Ž = 4 cm π‘‘β„Ž/𝑑𝑑= 1/(3πœ‹ Γ— (4)^2 ) π‘‘β„Ž/𝑑𝑑 =1/48πœ‹ Since height is in cm & time is in sec ∴ 𝒅𝒉/𝒅𝒕=𝟏/πŸ’πŸ–π… cm/s Hence, Height of the sand cone is increasing at the rate of 𝟏/πŸ’πŸ–π… cm/s

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.