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Ex 6.1, 3 - Radius of a circle is increasing uniformly at 3 cm/s - Finding rate of change

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  1. Chapter 6 Class 12 Application of Derivatives
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Ex 6.1,3 The radius of a circle is increasing uniformly at the rate of 3 cm/s. Find the rate at which the area of the circle is increasing when the radius is 10 cm. Let r be the radius of circle . & A be the Area of circle. Given that Radius of a circle is increasing at the rate of 3cm/s Thus, ๐‘‘๐‘Ÿ๏ทฎ๐‘‘๐‘ก๏ทฏ = 3cm /sec We need to find rate of change of area of circle w. r. t time when r = 10 cm i.e. we need to find ๐‘‘๐ด๏ทฎ๐‘‘๐‘ก๏ทฏ when r = 10 cm We know that Area of circle = ฯ€r2 A = ฯ€r2 Differentiate w.r.t time ๐‘‘๐ด๏ทฎ๐‘‘๐‘ก๏ทฏ = ๐‘‘ ๐œ‹๐‘Ÿ2๏ทฏ๏ทฎ๐‘‘๐‘ก๏ทฏ ๐‘‘๐ด๏ทฎ๐‘‘๐‘ก๏ทฏ = ฯ€ ๐‘‘ ๐‘Ÿ2๏ทฏ๏ทฎ๐‘‘๐‘ก๏ทฏ ๐‘‘๐ด๏ทฎ๐‘‘๐‘ก๏ทฏ = ฯ€ ๐‘‘ ๐‘Ÿ2๏ทฏ๏ทฎ๐‘‘๐‘ก๏ทฏ ร— ๐‘‘๐‘Ÿ๏ทฎ๐‘‘๐‘Ÿ๏ทฏ ๐‘‘๐ด๏ทฎ๐‘‘๐‘ก๏ทฏ = ฯ€ ๐‘‘ ๐‘Ÿ2๏ทฏ๏ทฎ๐‘‘๐‘Ÿ๏ทฏ ร— ๐‘‘๐‘Ÿ๏ทฎ๐‘‘๐‘ก๏ทฏ ๐‘‘๐ด๏ทฎ๐‘‘๐‘ก๏ทฏ = ฯ€. 2r . ๐‘‘๐‘Ÿ๏ทฎ๐‘‘๐‘ก๏ทฏ ๐‘‘๐ด๏ทฎ๐‘‘๐‘ก๏ทฏ = 2ฯ€r . ๐’…๐’“๏ทฎ๐’…๐’•๏ทฏ ๐‘‘๐ด๏ทฎ๐‘‘๐‘ก๏ทฏ = 2ฯ€r . 3 ๐‘‘๐ด๏ทฎ๐‘‘๐‘ก๏ทฏ = 6ฯ€r When ๐‘Ÿ = 10cm ๐‘‘๐ด๏ทฎ๐‘‘๐‘ก๏ทฏ๏ทฏ๏ทฎ๐‘Ÿ =10๏ทฏ = 6 ร— ฯ€ ร— 10 ๐‘‘๐ด๏ทฎ๐‘‘๐‘ก๏ทฏ๏ทฏ๏ทฎ๐‘Ÿ =10๏ทฏ = 60 ฯ€ Since area is in cm2 & time is in sec ๐‘‘๐ด๏ทฎ๐‘‘๐‘ก๏ทฏ = 60 ฯ€ ๐‘๐‘š2๏ทฎ๐‘ ๐‘’๐‘๏ทฏ ๐‘‘๐ด๏ทฎ๐‘‘๐‘ก๏ทฏ = 60 ฯ€ cm2/sec Hence Area is increasing at the rate of 60 ฯ€ cm2/sec when r = 10cm

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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.
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