# Ex 6.1,7

Last updated at Dec. 8, 2016 by Teachoo

Last updated at Dec. 8, 2016 by Teachoo

Transcript

Ex 6.1,7 The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When x = 8cm and y = 6cm, find the rates of change of (a) the perimeter, and (b) the area of the rectangle. Let length of rectangle = ๐ฅ & width of rectangle = ๐ฆ Given that Length of rectangle is decreasing at the rate of 5 cm/min i.e. ๐๐ฅ๏ทฎ๐๐ก๏ทฏ = โ 5 cm/ min And width of rectangle is increasing at the rate of 4 cm/min I.e. ๐๐ฆ๏ทฎ๐๐ก๏ทฏ = 4 cm/min Let p be the perimeter of rectangle We need to find rate of change of perimeter w. r. t time when ๐ฅ = 8 cm & y = 6 cm i.e. find ๐๐๏ทฎ๐๐ก๏ทฏ when ๐ฅ = 8 cm & ๐ฆ = 6 cm We know that Perimeter of rectangle = 2 (length + width) p = 2 (๐ฅ + ๐ฆ) Different w. r. t time ๐๐๏ทฎ๐๐ก๏ทฏ = ๐ 2 ๐ฅ+๐ฆ๏ทฏ๏ทฏ๏ทฎ๐๐ก๏ทฏ ๐๐๏ทฎ๐๐ก๏ทฏ = 2 ๐ (๐ฅ + ๐ฆ)๏ทฎ๐๐ก๏ทฏ ๐๐๏ทฎ๐๐ก๏ทฏ = 2. ๐๐ฅ๏ทฎ๐๐ก๏ทฏ + ๐๐ฆ๏ทฎ๐๐ก๏ทฏ๏ทฏ From (1) & (2) ๐๐๏ทฎ๐๐ก๏ทฏ = 2 ( โ 5 + 4 ) ๐๐๏ทฎ๐๐ก๏ทฏ = 2 ( โ 1) ๐๐๏ทฎ๐๐ก๏ทฏ = โ 2 Since perimeter is in cm & time is in sec. ๐๐๏ทฎ๐๐๏ทฏ = โ 2 ๐๐๏ทฎ๐ ๐๐๏ทฏ ๐๐๏ทฎ๐๐ก๏ทฏ = โ 2 cm/sec Hence, perimeter is decreasing at the rate of 2cm/sec when ๐ฅ = 8 cm & y = 6 cm (ii) Let A be the Area of rectangle we need to find rate of change of area w.r.t time when ๐ฅ = 8 & ๐ฆ = 6 cm i.e. ๐๐ด๏ทฎ๐๐ก๏ทฏ when ๐ฅ = 8 cm & ๐ฆ = 6 cm We know that Area of rectangle = length ร width A = ๐ฅ . ๐ฆ Differentiate w.r.t time ๐๐ด๏ทฎ๐๐ก๏ทฏ = ๐(๐ฅ . ๐ฆ)๏ทฎ๐๐ก๏ทฏ ๐๐ด๏ทฎ๐๐ก๏ทฏ = ๐ ๐ฅ๏ทฏ๏ทฎ๐๐ก๏ทฏ . ๐ฆ + ๐๐ฆ๏ทฎ๐๐ก๏ทฏ . ๐ฅ. ๐๐ด๏ทฎ๐๐ก๏ทฏ = โ 5 . ๐ฆ + 4 . ๐ฅ ๐๐ด๏ทฎ๐๐ก๏ทฏ = 4๐ฅ โ 5๐ฆ when ๐ฅ = 8 cm & ๐ฆ = 6 cm ๐๐ด๏ทฎ๐๐ก๏ทฏ = 4 (8) โ 5(6) ๐๐ด๏ทฎ๐๐ก๏ทฏ = 32 โ 30 ๐๐ด๏ทฎ๐๐ก๏ทฏ = 2 Since Area is in cm2 & time is in sec So ๐๐ด๏ทฎ๐๐ก๏ทฏ = 2๐๐2๏ทฎ๐ ๐๐๏ทฏ ๐๐ด๏ทฎ๐๐ก๏ทฏ = 2 cm2/ sec Hence, Area is increasing at the rate of 2cm2/sec when ๐ฅ = 80cm & ๐ฆ = 60 cm

Chapter 6 Class 12 Application of Derivatives

Serial order wise

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 7 years. He provides courses for Mathematics and Science from Class 6 to 12. You can learn personally from here https://www.teachoo.com/premium/maths-and-science-classes/.