Interpretation of the Derivative — Question 5

Question 5 - Solution

We are given: $$P(t) = 500 + 40t - 0.5t^2$$

(a) Rate of Change of Population:

The rate of change of the population is given by the derivative: $$P'(t) = \frac{d}{dt}(500 + 40t - 0.5t^2) = 40 - t$$

$$\boxed{P'(t) = 40 - t}$$

(b) Growth Rate in 2010:

Since $t$ is years since 2000, 2010 corresponds to $t = 10$.

$$P'(10) = 40 - 10 = 30$$

Interpretation: In 2010, the population was increasing at a rate of 30 thousand people per year.

$$\boxed{P'(10) = 30 \quad \text{(thousand people/year)}}$$

(c) When is the Population Growth Rate Zero?

Set $P'(t) = 0$: $$40 - t = 0 \Rightarrow t = 40$$

$$\boxed{t = 40}$$

(d) Interpretation:

Since $t = 40$ corresponds to the year $2000 + 40 = 2040$, the population stops growing in 2040.

Conclusion: In 2040, the population reaches its maximum and stops increasing. After that, it begins to decline.

$$\boxed{\text{The population stops growing in 2040.}}$$