Question

Suppose the population of a certain city is 5358 thousand. It is expected to decrease to 4565 thousand in 50 years. Find the percent decrease. The percent decrease is approximately nothing​%. ​(Round to the nearest​ tenth.)

Answer 1

Answer:

The population decreases at the rate of 0.32% a year.

Step-by-step explanation:

The population of this certain city can be modeled by this following differential equation.

$\frac{dP}{dt} = Pr$

where r is the growth rate(r>0 means that the population increases, r < 0 it decreases).

We can solve this by the variable separation method. We have that:

$\frac{dP}{P} = rdt$

Integrating both sides, we have

$ln{P} = rt + P(0)$

where P(0) is the initial population.

To find P in function of t, we apply the exponential to both sides.

$e^{ln{P}} = e^{rt + P(0)}$

$P(t) = P(0)e^{rt}$

The initial population of the city 5,358,000. So P(0) = 5,358,000.

It decreases to 4,565,000 in 50 years. So P(50) = 4,565,000.

Applying to the bold equation:

$5,358,000 = 4,565,000e^{50r}$

$e^{50r} = 1.174$

To find the growth rate, we apply ln to both sides.

$ln{e^{50r}} = ln{1.174}$

$50r = 0.16$

r = $\frac{0.16}{50} = 0.0032 = 0.32%$

The population decreases at the rate of 0.32% a year.