differential equations

Question

The population of a community is known to increase at a rate proportional to the number of people present at time t. If an initial population P₀

 has doubled in 6 years, how long will it take to triple? (Round your answer to one decimal place.)

___________ yr

How long will it take to quadruple? (Round your answer to one decimal place.)

___________ yr

Answer 1

Hi Destiny,

Thanks for your question! If you found my answer below helpful, I would greatly appreciate any review or feedback as I am new to the platform. (That said, I am a very experienced mathematician!

1. Doubling in 6 Years: The initial population P0 doubles in 6 years. So, 2 * P0 = P0 * exp(6 * k) Simplifying, exp(6 * k) = 2 Solving for k, k = ln(2) / 6 k is approximately 0.1155
2. Time to Triple: To find the time it takes to triple, set P(t) = 3 * P0 3 * P0 = P0 * exp(0.1155 * t) Simplifying, exp(0.1155 * t) = 3 Solve for t, t = ln(3) / 0.1155 t is approximately 10.6 years, rounded to one decimal place
3. Time to Quadruple: To find the time it takes to quadruple, set P(t) = 4 * P0 4 * P0 = P0 * exp(0.1155 * t) Simplifying, exp(0.1155 * t) = 4 Solve for t, t = ln(4) / 0.1155 t is approximately 14.2 years, rounded to one decimal place

Feel free to contact me if you would like to delve deeper into this or have any further questions!

Thank you,

Benjamin M.