1. **State the problem:** Given the population function: $$P(x) = 20 - \frac{6}{x+1}$$ where $x$ is the number of months from now (in thousands), find the rate of change of population with respect to time $x$ in months. 2. **Find the derivative of $P(x)$ to get the rate of change:** $$P(x) = 20 - 6(x+1)^{-1}$$ Using the power rule and chain rule: $$\frac{dP}{dx} = 0 - 6 \times (-1)(x+1)^{-2} = \frac{6}{(x+1)^2}$$ So, the rate of change of population is: $$\frac{dP}{dx} = \frac{6}{(x+1)^2}$$ (thousands per month) 3. **Evaluate the rate of change at $x=15$ months:** $$\frac{dP}{dx}\bigg|_{x=15} = \frac{6}{(15+1)^2} = \frac{6}{16^2} = \frac{6}{256} = \frac{3}{128} \approx 0.0234$$ Population changes about 0.0234 thousand or 23.4 people per month at 15 months from now. 4. **Calculate the actual population change during the 16th month:** Change during 16th month equals: $$P(16) - P(15) = \left(20 - \frac{6}{17}\right) - \left(20 - \frac{6}{16}\right) = -\frac{6}{17} + \frac{6}{16} = 6\left(\frac{1}{16} - \frac{1}{17}\right)$$ Calculate difference: $$\frac{1}{16} - \frac{1}{17} = \frac{17 - 16}{16 \times 17} = \frac{1}{272}$$ Hence change: $$6 \times \frac{1}{272} = \frac{6}{272} = \frac{3}{136} \approx 0.0221$$ Population increases by about 0.0221 thousand or 22.1 people during the 16th month. **Final answers:** $$\frac{dP}{dx} = \frac{6}{(x+1)^2}$$ Rate at 15 months: approximately 0.0234 thousand per month Population increase during 16th month: approximately 0.0221 thousand