Example 7:
It is estimated that $x$ months from now, the population of a certain community will be
$P(x)=x^2+20 x+8000$
(a) At what rate will the population be changing with respect to 15 months from now?
(b) By how much will the population actually change during the $16^{\text {th }}$ month?
Answer (1)
Find the derivative of the population function: P ′ ( x ) = 2 x + 20 .
Evaluate the derivative at x = 15 to find the rate of change at 15 months: P ′ ( 15 ) = 50 .
Calculate the population at the end of the 15th and 16th months: P ( 15 ) = 8525 and P ( 16 ) = 8576 .
Find the actual change in population during the 16th month: P ( 16 ) − P ( 15 ) = 51 .
The rate of change of population at 15 months is 50 people per month, and the actual change in population during the 16th month is 51 people.
Explanation
Problem Analysis We are given the population function P ( x ) = x 2 + 20 x + 8000 , where x represents the number of months from now. We need to find two things: (a) The rate of change of the population 15 months from now. (b) The actual change in population during the 16th month.
Finding the Derivative (a) To find the rate of change of the population, we need to find the derivative of the population function P ( x ) with respect to x . This will give us P ′ ( x ) , which represents the rate of change of the population at any time x .
The derivative of P ( x ) = x 2 + 20 x + 8000 is: P ′ ( x ) = d x d ( x 2 + 20 x + 8000 ) = 2 x + 20
Rate of Change at 15 Months Now, we need to evaluate P ′ ( x ) at x = 15 to find the rate of change of the population 15 months from now: P ′ ( 15 ) = 2 ( 15 ) + 20 = 30 + 20 = 50 So, the population will be changing at a rate of 50 people per month 15 months from now.
Population at 15 Months (b) To find the actual change in population during the 16th month, we need to calculate the population at the end of the 15th month, P ( 15 ) , and at the end of the 16th month, P ( 16 ) . Then, we subtract P ( 15 ) from P ( 16 ) to find the difference.
First, let's find P ( 15 ) :
P ( 15 ) = ( 15 ) 2 + 20 ( 15 ) + 8000 = 225 + 300 + 8000 = 8525
Population at 16 Months Next, let's find P ( 16 ) :
P ( 16 ) = ( 16 ) 2 + 20 ( 16 ) + 8000 = 256 + 320 + 8000 = 8576
Population Change During 16th Month Now, we subtract P ( 15 ) from P ( 16 ) to find the actual change in population during the 16th month: Change = P ( 16 ) − P ( 15 ) = 8576 − 8525 = 51 So, the population will actually change by 51 people during the 16th month.
Final Answer (a) The rate at which the population will be changing 15 months from now is 50 people per month. (b) The population will actually change by 51 people during the 16th month.
Examples
Understanding population growth rates is crucial for urban planning. For instance, knowing how quickly a city's population is expanding helps officials allocate resources effectively, such as building new schools, hospitals, and transportation infrastructure to accommodate the growing number of residents. This problem demonstrates how mathematical models can provide valuable insights into demographic trends, enabling proactive decision-making and sustainable development.
