We are given the population of a city as a function of time, $P(t) = (0.5t - 6)(0.3t + 9) + 90$, where $t$ is the number of years from now, and we are asked to find how fast the population will be growing after 22 years.

To find the rate at which the population is growing, we need to calculate the derivative of $P(t)$ with respect to $t$, denoted as $P'(t)$. This derivative will give us the rate of change of the population at any given time.

Step 1: Differentiate $P(t)$

The function $P(t)$ is composed of a product of two functions, so we will use the product rule to differentiate it.

The product rule states that: $\frac{d}{dt} [f(t)g(t)] = f'(t)g(t) + f(t)g'(t)$ Let: $f(t) = 0.5t - 6 \quad \text{and} \quad g(t) = 0.3t + 9$

Now, differentiate both $f(t)$ and $g(t)$: $f'(t) = 0.5 \quad \text{and} \quad g'(t) = 0.3$

Applying the product rule: $P'(t) = f'(t)g(t) + f(t)g'(t)$ Substituting the values of $f(t), f'(t), g(t),$ and $g'(t)$: $P'(t) = (0.5)(0.3t + 9) + (0.5t - 6)(0.3)$

Step 2: Simplify the expression

Simplifying each term: $P'(t) = 0.15t + 4.5 + 0.15t - 1.8$ $P'(t) = 0.3t + 2.7$

Step 3: Evaluate $P'(t)$ at $t = 22$

Now, substitute $t = 22$ into the derivative: $P'(22) = 0.3(22) + 2.7 = 6.6 + 2.7 = 9.3$

So, the population will be growing at a rate of 9.3 thousand people per year after 22 years.

Final Answer:

The population will be growing at a rate of 9.30 thousand people per year after 22 years.

Would you like more details on this or have any other questions?

Here are 5 related questions for further exploration:

1. How would the result change if the population function had an additional exponential term?
2. What is the significance of the constant $+90$ in the population function?
3. What is the population of the city exactly 22 years from now?
4. How does the growth rate change as $t$ increases indefinitely?
5. How could we determine when the population growth will stop?

Tip: The derivative of a function provides the rate of change, which is crucial in understanding how things evolve over time in real-world applications like population growth.