Last updated at April 19, 2021 by Teachoo

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 ๐ be the radius & ๐ be height of the sand cone & V be the volume of cone Also, Sand is pouring from a pipe at the rate of 12๐๐^3/sec i.e. Rate of volume of a cone w.r.t time is 12๐๐^3/sec i.e. ๐ ๐ฝ/๐ ๐ ="12" ๐๐^๐ "/sec" We need to find how fast height of the Cone is increasing when height is 4cm i.e. find ๐ ๐/๐ ๐ when ๐=๐๐๐ 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. โ=1/6 ๐ 6โ=๐ ๐=๐๐ We know that Volume of a cone = 1/3 ๐(๐^2 )โ V = 1/3 ฯใ ๐ใ^2 โ V = 1/3 ฯ (๐๐)^2 โ V = 1/3 ฯ ร 36โ^2 รโ V = 1/3 ฯ ร 36 โ^3 V = 12ฯ ๐^๐ (โ("From (2)" : ๐" = 6" โ)) Differentiating w.r.t ๐ก ๐๐/๐๐ก=๐(12๐โ^3 )/๐๐ก ๐๐/๐๐ก=12๐ ๐(โ^3 )/๐๐ก ๐๐/๐๐ก=12๐ ร๐(โ^3 )/๐๐ก ร ๐โ/๐โ ๐๐/๐๐ก=12๐ ร๐ (๐^๐ )/๐ ๐ ร ๐โ/๐๐ก ๐ ๐ฝ/๐ ๐=12๐ ร 3โ^2 ร ๐โ/๐๐ก ๐๐=12๐ ร 3โ^2 ร ๐โ/๐๐ก 12/(12๐ ร 3โ^2 )=๐โ/๐๐ก ๐ ๐/๐ ๐=๐/(๐๐ ๐^๐ ) (From (1): ๐ ๐ฝ/๐ ๐ ="12" ) Putting โ = 4 cm ๐โ/๐๐ก= 1/(3๐ ร (4)^2 ) ๐โ/๐๐ก =1/48๐ Since height is in cm & time is in sec โด ๐ ๐/๐ ๐=๐/๐๐๐ cm/s Hence, Height of the sand cone is increasing at the rate of ๐/๐๐๐ cm/s

Ex 6.1

Ex 6.1, 1
Deleted for CBSE Board 2021 Exams only

Ex 6.1,2 Deleted for CBSE Board 2021 Exams only

Ex 6.1,3 Deleted for CBSE Board 2021 Exams only

Ex 6.1,4 Important Deleted for CBSE Board 2021 Exams only

Ex 6.1,5 Important Deleted for CBSE Board 2021 Exams only

Ex 6.1,6 Deleted for CBSE Board 2021 Exams only

Ex 6.1,7 Deleted for CBSE Board 2021 Exams only

Ex 6.1,8 Deleted for CBSE Board 2021 Exams only

Ex 6.1,9 Deleted for CBSE Board 2021 Exams only

Ex 6.1,10 Important Deleted for CBSE Board 2021 Exams only

Ex 6.1,11 Important Deleted for CBSE Board 2021 Exams only

Ex 6.1,12 Deleted for CBSE Board 2021 Exams only

Ex 6.1,13 Important Deleted for CBSE Board 2021 Exams only

Ex 6.1,14 Deleted for CBSE Board 2021 Exams only You are here

Ex 6.1,15 Important Deleted for CBSE Board 2021 Exams only

Ex 6.1,16 Deleted for CBSE Board 2021 Exams only

Ex 6.1,17 Deleted for CBSE Board 2021 Exams only

Ex 6.1, 18 Deleted for CBSE Board 2021 Exams only

Chapter 6 Class 12 Application of Derivatives

Serial order wise

About the Author

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.