# Ex 6.1,8

Last updated at Dec. 8, 2016 by Teachoo

Last updated at Dec. 8, 2016 by Teachoo

Transcript

Ex 6.1,8 A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimeters of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm. Balloon is spherical Let r be the radius of spherical balloon . & v be the volume of spherical balloon. It is given that The balloon is inflated by pumping in in 900 cubic cm of gas per sec i.e. volume of balloon increasing at the rate of 9000 cm3/sec i.e. ๐๐ฃ๏ทฎ๐๐ก๏ทฏ = 900 cm3/sec We need to calculate the rate of change at which radius of Balloon increases when radius is 15 cm i.e. we need to calculate ๐๐๏ทฎ๐๐ก๏ทฏ when r = 15 cm we know that Volume of sphere = 4๏ทฎ3๏ทฏ ฯr3 i.e. V = 4๏ทฎ3๏ทฏ ฯr3 Differentiate w.r.t time(๐ก) ๐๐ฃ๏ทฎ๐๐ก๏ทฏ = ๐ 4๏ทฎ3๏ทฏ ๐๐3๏ทฏ๏ทฎ๐๐ก๏ทฏ ๐๐ฃ๏ทฎ๐๐ก๏ทฏ = 4๏ทฎ3๏ทฏ ฯ ๐ (๐3)๏ทฎ๐๐ก๏ทฏ 900 = 4๏ทฎ3๏ทฏ ฯ . ๐(๐3)๏ทฎ๐๐ก๏ทฏ ร ๐๐๏ทฎ๐๐๏ทฏ 900 = 4๏ทฎ3๏ทฏ ฯ . ๐(๐3)๏ทฎ๐๐ก๏ทฏ ร ๐๐๏ทฎ๐๐ก๏ทฏ 900 = 4๏ทฎ3๏ทฏ ฯ . 3r2 . ๐๐๏ทฎ๐๐ก๏ทฏ 900 = 4 ร ฯ ร r2 . ๐๐๏ทฎ๐๐ก๏ทฏ 900๏ทฎ4 ร ๐ ร๐2๏ทฏ = ๐๐๏ทฎ๐๐ก๏ทฏ ๐๐๏ทฎ๐๐ก๏ทฏ = 900 ร 7๏ทฎ2 ร22 ร๐2๏ทฏ We need to find ๐๐๏ทฎ๐๐ก๏ทฏ at r = 15 cm ๐๐๏ทฎ๐๐ก๏ทฏ = 900๏ทฎ4 ร๐ ร 15๏ทฏ๏ทฎ2๏ทฏ๏ทฏ ๐๐๏ทฎ๐๐ก๏ทฏ = 1๏ทฎ๐๏ทฏ Since radius in cm & time in sec So, ๐๐๏ทฎ๐๐ก๏ทฏ = ๐ ๐๐๏ทฎ๐ ๐๐๐๏ทฏ Hence ,the radius of the balloon is increasing at the rate of 1๏ทฎ๐๏ทฏ cm/sec when r = 15 cm.

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.