web analytics

Example 14 - In a bank, principal increases 5%. In how many - Variable separation - Statement given

Slide4.JPG
Slide5.JPG Slide6.JPG

  1. Chapter 9 Class 12 Differential Equations
  2. Serial order wise
Ask Download

Transcript

Example 14 In a bank, principal increases continuously at the rate of 5% per year. In how many years Rs 1000 double itself ? Rate of change of principal = 5% of principal 𝑑𝑃﷮𝑑𝑡﷯ = 5% × P 𝑑𝑃﷮𝑑𝑡﷯ = 5﷮100﷯ × P 𝑑𝑃﷮𝑑𝑡﷯ = 1﷮20﷯ × P 𝑑𝑝﷮𝑃﷯ = 𝑑𝑡﷮20﷯ Integrating both sides ﷮﷮ 𝑑𝑝﷮𝑃﷯﷯ = ﷮﷮ 𝑑𝑡﷮20﷯﷯ log 𝑃﷯ = 𝑡﷮20﷯ + C Removing log P = e﷮ 𝑡﷮20﷯﷯ × ec P = K e﷮ 𝑡﷮20﷯﷯ where k = ec Now, we have to find in how many years Rs 1000 double it self We have to find time? Now, At t = 0, P = 1000 Putting in (1) P = k e﷮ 𝑡﷮20﷯﷯ 1000 = k e﷮ 0﷮20﷯﷯ 1000 = k e0 1000 = k × 1 1000 = k So k = 1000 Put k = 1000 in (1) i.e, P = ke﷮ 𝑡﷮20﷯﷯ P = 1000 e﷮ 𝑡﷮20﷯﷯ Now at t = t, P = 2000 2000 = 1000 e﷮ 𝑡﷮20﷯﷯ 2000﷮1000﷯ = e﷮ 𝑡﷮20﷯﷯ 2 = e﷮ 𝑡﷮20﷯﷯ e﷮ 𝑡﷮20﷯﷯ = 2 Taking log both sides log e﷮ 𝑡﷮20﷯﷯﷯ = log 2 𝑡﷮20﷯. log e = log 2 𝑡﷮20﷯ × 1 = log 2 t = 20 log 2

About the Author

Davneet Singh's photo - Teacher, Computer Engineer, Marketer
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
Jail