Ex 6.1, 5 - A stone is dropped into a quiet lake and waves move - Ex 6.1


  1. Chapter 6 Class 12 Application of Derivatives
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Ex 6.1,5 A stone is dropped into a quiet lake and waves move in circles at the speed of 5 cm/s. At the instant when the radius of the circular wave is 8 cm, how fast is the enclosed area increasing? Let r be the radius of circle & A be the Area of circle When stone is dropped into a lake waves move in a circle at the speed of 5 cm/sec Thus, ๐‘‘๐‘Ÿ๏ทฎ๐‘‘๐‘ก๏ทฏ = 5cm/sec We need find how fast area increasing w. r. t time when radius is 8cm i.e. we need to find ๐‘‘๐ด๏ทฎ๐‘‘๐‘ก๏ทฏ when r = 8 cm. We know that Area of circle = ฯ€r2 A = ฯ€r2 Different w. r. t time ๐‘‘๐ด๏ทฎ๐‘‘๐‘ก๏ทฏ = ๐‘‘(๐œ‹๐‘Ÿ2)๏ทฎ๐‘‘๐‘ก๏ทฏ ๐‘‘๐ด๏ทฎ๐‘‘๐‘ก๏ทฏ = ฯ€ ๐‘‘(๐‘Ÿ2)๏ทฎ๐‘‘๐‘ก๏ทฏ ๐‘‘๐ด๏ทฎ๐‘‘๐‘ก๏ทฏ = ฯ€ ๐‘‘(๐‘Ÿ2)๏ทฎ๐‘‘๐‘ก๏ทฏ ร— ๐‘‘๐‘Ÿ๏ทฎ๐‘‘๐‘Ÿ๏ทฏ ๐‘‘๐ด๏ทฎ๐‘‘๐‘ก๏ทฏ = ฯ€ ๐‘‘(๐‘Ÿ2)๏ทฎ๐‘‘๐‘Ÿ๏ทฏ ร— ๐‘‘๐‘Ÿ๏ทฎ๐‘‘๐‘ก๏ทฏ ๐‘‘๐ด๏ทฎ๐‘‘๐‘ก๏ทฏ = ฯ€ ร— 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 . 5 ๐‘‘๐ด๏ทฎ๐‘‘๐‘ก๏ทฏ = 10ฯ€r When r = 8 cm ๐‘‘๐ด๏ทฎ๐‘‘๐‘ก๏ทฏ๏ทฏ๏ทฎ๐‘Ÿ = 8๏ทฏ = 10 ร— ฯ€ ร— 8 ๐‘‘๐ด๏ทฎ๐‘‘๐‘ก๏ทฏ๏ทฏ๏ทฎ๐‘Ÿ = 8๏ทฏ = 80ฯ€ Since Area is in cm2 & time is in sec ๐‘‘๐ด๏ทฎ๐‘‘๐‘ก๏ทฏ๏ทฏ๏ทฎ๐‘Ÿ = 8๏ทฏ= 80ฯ€ ๐‘๐‘š2๏ทฎ๐‘ ๐‘’๐‘๏ทฏ ๐‘‘๐ด๏ทฎ๐‘‘๐‘ก๏ทฏ๏ทฏ๏ทฎ๐‘Ÿ = 8๏ทฏ= 80ฯ€ cm2/sec Hence Area is increasing at the rate of 80ฯ€ cm2/sec when r = 8 cm

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