Understanding Population Growth in a Small Town

In this article, we will investigate the growth of a small town whose initial population is 500. The population is modeled by the function P(t) 500e0.2t, where P(t) is the population after t years. Our objective is to show that the ratio of the rate of change of the population P'(t) to the population P(t) is constant. Let's explore this in detail.

Finding the Derivative of the Population Function

The population function is given by P(t) 500e0.2t. To find the rate of change of the population, we need to find its derivative with respect to time t.

Using the chain rule, we differentiate the function:

P'(t) d/dt[500e0.2t] 500 * 0.2e0.2t 100e0.2t

This tells us that the rate of change of the population at any time t is given by 100e0.2t.

Calculating the Ratio of the Rate of Change to the Population

To show that the ratio of the rate of change of the population P'(t) to the population P(t) is constant, we need to compute the ratio:

Ratio P'(t) / P(t) (100e0.2t) / (500e0.2t)

Simplifying this expression, we obtain:

Ratio 100/500 1/5

Thus, the ratio of the rate of change of the population to the population itself is

1/5

This value is clearly constant and does not depend on t.

Conclusion

The ratio of the rate of change of the population to the population is shown to be a constant, which is 1/5. This indicates that the growth rate of the population is consistently one-fifth of the population at any given time.

Now, let's consider the rate of change starting at t0. The population at any time t can be expressed as:

P(t) P(t0)e0.2(t - t0)

The rate of change at any time t can be written as:

P'(t) 0.2P(t) 0.2P(t0)e0.2(t - t0)

The ratio of the rate of change at time t to the rate of change at time t0 is:

(P'(t) / P'(t0)) (0.2P(t0)e0.2(t - t0)) / (0.2P(t0)e0.2(t0 - t0)) e0.2(t - t0)

This confirms that the rate of change starting at time t0 is a constant multiple of that starting at time t0, and this constant is e0.2(t - t0), which itself is always a constant for any fixed t0.

Conclusion

In summary, the ratio of the rate of change of the population to the population itself is indeed constant, as shown by the calculations. This constant value of 1/5 indicates a consistent growth rate of the population, highlighting the nature of exponential growth in the context of population dynamics.

For further exploration, readers can apply similar analysis to other scenarios of exponential growth and monitor the behavior of the ratio under different conditions.