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Example 35 A circular disc of radius 3 cm is being heated. Due to expansion, its radius increases at the rate of 0.05 cm/s. Find the rate at which its area is increasing when radius is 3.2 cm.Let r be the radius of circle . & A be the Area of circle. Given that Radius increases at the rate of 0.05 cm/s Thus, ๐’…๐’“/๐’…๐’• = 0.05 cm /sec We need to find rate of change of area of circle w. r. t time when r = 3.2 cm i.e. we need to find ๐’…๐‘จ/๐’…๐’• when r = 3.2 cm We know that Area of circle = ฯ€r2 A = ฯ€r2 Differentiating w.r.t time ๐’…๐‘จ/๐’…๐’• = ๐’…(๐…๐’“๐Ÿ)/๐’…๐’• ๐‘‘๐ด/๐‘‘๐‘ก = ฯ€ ๐‘‘(๐‘Ÿ2)/๐‘‘๐‘ก ๐‘‘๐ด/๐‘‘๐‘ก = ฯ€ ๐‘‘(๐‘Ÿ2)/๐‘‘๐‘ก ร— ๐’…๐’“/๐’…๐’“ ๐‘‘๐ด/๐‘‘๐‘ก = ฯ€ ๐’…(๐’“๐Ÿ)/๐’…๐’“ ร— ๐‘‘๐‘Ÿ/๐‘‘๐‘ก ๐‘‘๐ด/๐‘‘๐‘ก = ฯ€. 2r . ๐‘‘๐‘Ÿ/๐‘‘๐‘ก ๐‘‘๐ด/๐‘‘๐‘ก = 2ฯ€r . ๐’…๐’“/๐’…๐’• ๐‘‘๐ด/๐‘‘๐‘ก = 2ฯ€r . 0.05 ๐‘‘๐ด/๐‘‘๐‘ก = 0.1 ร— ฯ€r When ๐’“ = 3.2 cm โ”œ ๐‘‘๐ด/๐‘‘๐‘กโ”ค|_(๐‘Ÿ =10) = 0.1 ร— ฯ€ ร— 3.2 โ”œ ๐‘‘๐ด/๐‘‘๐‘กโ”ค|_(๐‘Ÿ =10) = 0.320ฯ€ Since area is in cm2 & time is in seconds ๐’…๐‘จ/๐’…๐’• = 0.320ฯ€ cm2/s Hence, Area is increasing at the rate of 0.320ฯ€ cm2/s when r = 0.32 cm

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.