Math Problem Statement

A population grows according to an exponential growth model. The initial population is Po=4 , and the growth rate is r=0.4

Solution

The population grows according to the exponential growth model, which is typically given by the formula:

P(t)=P0ertP(t) = P_0 e^{rt}

Where:

  • P(t)P(t) is the population at time tt,
  • P0P_0 is the initial population,
  • rr is the growth rate (per unit time),
  • tt is the time that has passed,
  • ee is Euler's number (approximately 2.71828).

Given the initial population P0=4P_0 = 4 and the growth rate r=0.4r = 0.4, the population at any time tt is modeled by the equation:

P(t)=4e0.4tP(t) = 4 e^{0.4t}

Example: Population after 5 years

If you want to calculate the population after 5 years, substitute t=5t = 5 into the equation:

P(5)=4e0.4×5=4e24×7.3891=29.5564P(5) = 4 e^{0.4 \times 5} = 4 e^{2} \approx 4 \times 7.3891 = 29.5564

So, after 5 years, the population would be approximately 29.56.

Would you like to calculate the population for a specific time or explore related details?

5 Related Questions:

  1. What will the population be after 10 years?
  2. How long will it take for the population to double?
  3. Can you solve for the time when the population reaches 100?
  4. How does changing the growth rate rr affect the population over time?
  5. What would happen to the population if it started at a different value of P0P_0?

Tip:

When solving exponential growth problems, it's helpful to memorize the doubling time formula: Tdouble=ln(2)rT_{\text{double}} = \frac{\ln(2)}{r}.

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Math Problem Analysis

Mathematical Concepts

Exponential Growth
Population Growth Models

Formulas

P(t) = P_0 * e^(rt)
Doubling Time Formula: T_double = ln(2)/r

Theorems

Exponential Growth Theorem

Suitable Grade Level

Grades 10-12