Ex 9.4, 20 - Class 12

Transcript

Ex 9.4, 20 In a bank, principal increases continuously at the rate of 𝑟% per year. Find the value of r if Rs 100 double itself in 10 years log﷮2=0.6931﷯﷯ Let Principle = p Given, principle increases ar rate r % per year ∴ 𝑑𝑝﷮𝑑𝑡﷯ = 𝑟% × P ∴ 𝑑𝑝﷮𝑑𝑡﷯ = 𝑟﷮100﷯ × p 𝑑𝑝﷮𝑝﷯ = 𝑟﷮100﷯ dt Integrating both sides ﷮﷮ 𝑑𝑝﷮𝑝﷯﷯ = 𝑟﷮100﷯ ﷮﷮𝑑𝑡﷯ log p = 𝑟𝑡﷮100﷯ + log c log p − log c = 𝑟𝑡﷮100﷯ log 𝑝﷮𝑐﷯ = 𝑟𝑡﷮100﷯ 𝑝﷮𝑐﷯ = e﷮ 𝑟𝑡﷮100﷯﷯ Initially t = 0 and p = 100 100﷮𝑐﷯ = e﷮ 𝑟.0﷮100﷯﷯ 100﷮𝑐﷯ = e﷮0﷯ 100﷮𝑐﷯ = 1 c = 100 Put this in equation (1) 𝑝﷮100 ﷯ = e﷮ 𝑟𝑡﷮100﷯﷯ Also, At t = 10, p = 200 200﷮100 ﷯ = e﷮ 10𝑟﷮100﷯﷯ 2 = e﷮ 𝑟﷮10﷯﷯ log 2 = 𝑟﷮10﷯ 0.6931 = 𝑟﷮10﷯ r = 6.931 ∴ Rate of interest = r = 6.931 %