This is a population growth problem involving exponential growth since the population grows at a rate proportional to its current size. Let's solve it step-by-step.

Problem:

The population $P(t)$ at any time $t$ grows proportionally to the population present, modeled by: $\frac{dP}{dt} = kP,$ where $k$ is the proportional growth constant.

Given:

1. Initial population $P(0) = 500$,
2. The population increases by 15% in 10 years ($P(10) = 1.15 \cdot 500 = 575$),
3. We are asked to:
   • Find $P(30)$,
   • Find the rate of population growth $\frac{dP}{dt}$ at $t = 30$.

Step 1: General solution of the differential equation

The equation $\frac{dP}{dt} = kP$ solves to: $P(t) = P(0)e^{kt}.$

Step 2: Determine $k$ using given information

Using $P(10) = 575$: $P(10) = P(0)e^{10k}.$ Substitute $P(0) = 500$ and $P(10) = 575$: $575 = 500e^{10k}.$ Divide both sides by 500: $1.15 = e^{10k}.$ Take the natural logarithm of both sides: $\ln(1.15) = 10k.$ Solve for $k$: $k = \frac{\ln(1.15)}{10}.$

Using $\ln(1.15) \approx 0.13976$: $k \approx \frac{0.13976}{10} = 0.013976.$

Step 3: Find $P(30)$

The population at $t = 30$ is: $P(30) = P(0)e^{30k}.$ Substitute $P(0) = 500$ and $k \approx 0.013976$: $P(30) = 500e^{30 \cdot 0.013976}.$ Simplify the exponent: $30 \cdot 0.013976 = 0.41928.$ Now calculate: $P(30) = 500e^{0.41928}.$ Using $e^{0.41928} \approx 1.5202$: $P(30) \approx 500 \cdot 1.5202 = 760.1.$ Thus, the population in 30 years is approximately: $P(30) = 760.$

Step 4: Find the rate of growth at $t = 30$

The rate of growth is given by: $\frac{dP}{dt} = kP(t).$ Substitute $k \approx 0.013976$ and $P(30) \approx 760$: $\frac{dP}{dt} \bigg|_{t=30} = 0.013976 \cdot 760.$ Simplify: $\frac{dP}{dt} \bigg|_{t=30} \approx 10.62.$

Thus, the rate of growth at $t = 30$ is approximately: $\frac{dP}{dt} = 10.62 \, \text{individuals per year}.$

Final Answers:

1. Population at $t = 30$: $P(30) \approx 760$,
2. Rate of growth at $t = 30$: $\frac{dP}{dt} \approx 10.62 \, \text{individuals/year}$.

Do you want more details or clarification?
Here are 5 related questions to expand your understanding:

1. What happens to the population as $t \to \infty$?
2. How would the solution change if the initial population was different?
3. How do you derive the exponential growth equation step-by-step?
4. Can you explain how logarithms help in solving for $k$?
5. How does the proportional constant $k$ affect the growth rate?

Tip: When working with exponential growth, always remember that the proportional constant $k$ determines how quickly the population grows!