Ex 6.1,1 - Chapter 6 Class 12 Application of Derivatives

Transcript

Ex 6.1,1 Find the rate of change of the area of a circle with respect to its radius r when (a) r = 3 cm (b) r = 4 cm Radius of circle = 𝑟 & let A be the area of circle We need to find rate of change of Area w. r. t Radius i.e. we need to calculate 𝑑𝐴﷮𝑑𝑟﷯ We know that Area of Circle = 𝜋𝑟﷮2﷯ 𝑑𝐴﷮𝑑𝑟﷯ = 𝑑 𝜋𝑟﷮2﷯﷯﷮𝑑𝑟﷯ 𝑑𝐴﷮𝑑𝑟﷯ = 𝜋 𝑑 (𝑟﷮2﷯)﷮𝑑𝑟﷯﷯ 𝑑𝐴﷮𝑑𝑟﷯ = 𝜋 2𝑟﷯ 𝑑𝐴﷮𝑑𝑟﷯ = 2𝑟 𝜋 • when r = 3 cm 𝑑𝐴﷮𝑑𝑟﷯ = 2rπ Putting r = 3 cm 𝑑𝐴﷮𝑑𝑟﷯﷯﷮𝑟 = 3﷯= 2(3) π 𝑑𝐴﷮𝑑𝑟﷯﷯﷮𝑟 = 3﷯ = 6 π Since Area is in cm2 & radius is in cm 𝑑𝐴﷮𝑑𝑟﷯ = 6π 𝒄𝒎𝟐﷮𝒄𝒎﷯ Hence Area is increasing at the rate of 6π cm2/ cm when r = 3 cm (ii) r = 4 cm 𝑑𝐴﷮𝑑𝑟﷯ = 2rπ Putting r = 4 cm 𝑑𝐴﷮𝑑𝑟﷯ = 2(4)π 𝑑𝐴﷮𝑑𝑟﷯ = 8 π Since Area is in cm2 & radius is in cm 𝒅𝑨﷮𝒅𝒓﷯ = 8π 𝒄𝒎𝟐﷮𝒄𝒎﷯ Hence Area is increasing at the rate of 8π cm2/cm when r = 4 cm