The sides of an equilateral triangle are increasing at the rate of 2 cm/sec. The rate at which the area increases, when side is 10 cm is:

(A)10 cm 2 /s                    (B) 3 cm 2 /s

(C) 10 3 cm 2 /s               (D) 10/3 cm 2 /s

 

This question is similar to Ex 6.1, 1 - Chapter 6 Class 12 - Application of Derivatives

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  1. Chapter 6 Class 12 Application of Derivatives (Term 1)
  2. Serial order wise

Transcript

Question 6 The sides of an equilateral triangle are increasing at the rate of 2 cm/sec. The rate at which the area increases, when side is 10 cm is: 10 cm2/s (B) 3 cm2/s (C) 10โˆš๐Ÿ‘ cm2/s (D) 10/3 cm2/s Let Area of equilateral triangle = A cm2 & let Side = ๐’™ cm Given that Sides of equilateral triangle are increasing at the rate of 2 cm/sec โˆด ๐’…๐’™/๐’…๐’• = 2 We need to find rate of change of area w.r.t. side i.e., we need to find ๐’…๐‘จ/๐’…๐’• We know that Area of equilateral triangle = A = โˆš3/4 ๐‘ฅ^2 Finding rate of change of area Differentiating A w.r.t.x ๐‘‘๐ด/๐‘‘๐‘ก = โˆš3/4 (๐‘ฅ^2 )โ€ฒ ๐‘‘๐ด/๐‘‘๐‘ก = โˆš3/4 ร— (๐‘‘ใ€–(๐‘ฅใ€—^2))/๐‘‘๐‘ฅ ร— ๐‘‘๐‘ฅ/๐‘‘๐‘ก ๐‘‘๐ด/๐‘‘๐‘ก = โˆš3/4 (2๐‘ฅ) ๐‘‘๐‘ฅ/๐‘‘๐‘ก ๐’…๐‘จ/๐’…๐’• = (โˆš๐Ÿ‘ ๐’™)/๐Ÿ ๐’…๐’™/๐’…๐’• Putting ๐’…๐’™/๐’…๐’• = 2, from equation (1) ๐‘‘๐ด/๐‘‘๐‘ก = (โˆš3 ๐‘ฅ)/2 ร— 2 ๐‘‘๐ด/๐‘‘๐‘ก = โˆš๐Ÿ‘ ๐’™ Since, we have to find rate of change of area when side is 10 cm โˆด Putting ๐’™ = 10 cm in ๐‘‘๐ด/๐‘‘๐‘ก ๐’…๐‘จ/๐’…๐’• = 10 โˆš๐Ÿ‘ cm2/sec Hence, area increases at the rate of 10 โˆš๐Ÿ‘ cm2/sec So, the correct answer is (C) ๐’…๐‘จ/๐’…๐’• = (โˆš๐Ÿ‘ ๐’™)/๐Ÿ ๐’…๐’™/๐’…๐’• Putting ๐’…๐’™/๐’…๐’• = 2, from equation (1) ๐‘‘๐ด/๐‘‘๐‘ก = (โˆš3 ๐‘ฅ)/2 ร— 2 ๐‘‘๐ด/๐‘‘๐‘ก = โˆš๐Ÿ‘ ๐’™ Since, we have to find rate of change of area when side is 10 cm โˆด Putting ๐’™ = 10 cm in ๐‘‘๐ด/๐‘‘๐‘ก ๐’…๐‘จ/๐’…๐’• = 10 โˆš๐Ÿ‘ cm2/sec Hence, area increases at the rate of 10 โˆš๐Ÿ‘ cm2/sec So, the correct answer is (C)

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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.