Last updated at May 6, 2021 by Teachoo

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Example 4 The length x of a rectangle is decreasing at the rate of 3 cm/minute and the width y is increasing at the rate of 2 cm/minute. When x = 10 cm and y = 6 cm, find the rates of change of (a) the perimeter and (b) the area of the rectangle.Let Length of rectangle = ๐ฅ cm & Width of rectangle = ๐ฆ cm Given length ๐ฅ is decreasing at the rate of 3 cm/minute ๐ ๐/๐ ๐ = โ 3 cm/ min and width y is increasing at the rate of 2 cm/min ๐ ๐/๐ ๐ = 2 cm /min (i) Finding rate of change of Perimeter Let P be the perimeter of rectangle. We need to calculate rate of change of perimeter when ๐ฅ = 10 cm & ๐ฆ = 6 cm i.e. we need to calculate ๐๐/๐๐ก when ๐ฅ = 10 cm & ๐ฆ = 6 cm We know that Perimeter of rectangle = 2 (Length + Width) P = 2 (๐ฅ + ๐ฆ) Now ๐๐/๐๐ก= (๐ (2 (๐ฅ + ๐ฆ) ) )/๐๐ก ๐๐/๐๐ก= 2 [๐(๐ฅ + ๐ฆ)/๐๐ก] ๐ ๐ท/๐ ๐= 2 [๐ ๐/๐ ๐+ ๐ ๐/๐ ๐] From (1) & (2) ๐๐ฅ/๐๐ก = โ3 & ๐๐ฆ/๐๐ก = 2 ๐๐/๐๐ก= 2(โ 3 + 2) ๐๐/๐๐ก= 2 (โ1) ๐ ๐ท/๐ ๐= โ2 Since perimeter is in cm & time is in minute ๐๐/๐๐ก = โ 2 cm/min Therefore, perimeter is decreasing at the rate of 2 cm/min(ii) Finding rate of change of Area Let A be the Area of rectangle. We need to calculate Rate of change of area when ๐ฅ = 10cm & ๐ฆ = 6 cm i.e. we need to calculate ๐ ๐จ/๐ ๐ when ๐ฅ=10 & ๐ฆ=6 cm We know that Area of rectangle = Length ร Width A = ๐ฅ ร ๐ฆ Now, ๐๐ด/๐๐ก = (๐ (๐ฅ๐ฆ))/๐๐ก ๐๐ด/๐๐ก = ๐๐ฅ/๐๐ก ๐ฆ + ๐๐ฆ/๐๐ก ๐ฅ From (1) & (2) ๐๐ฅ/๐๐ก = โ3 & ๐๐ฆ/๐๐ก = 2 dA/dt = (โ3)๐ฆ+2 (๐ฅ) ๐ ๐จ/๐ ๐ = โ ๐๐+๐๐ Putting ๐ = 10 & ๐ = 6 cm ๐๐ด/๐๐ก = โ 3(6) + 2(10) = โ 18 + 20 = 2 Since Area is in cm2 & time is in time in minute ๐๐ด/๐๐ก = 2 cm2/min Hence, Area is increasing at the rate of 2 cm2/min.

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Chapter 6 Class 12 Application of Derivatives (Term 1)

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About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.