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 ) = x 2 + 20 x + 8000 , which is P ′ ( x ) = 2 x + 20 .
Evaluate P ′ ( 15 ) to find the rate of change at 15 months: P ′ ( 15 ) = 50 .
Calculate P ( 15 ) and P ( 16 ) to find the population at the end of the 15th and 16th months, respectively: P ( 15 ) = 8525 and P ( 16 ) = 8576 .
Determine the actual change in population during the 16th month: P ( 16 ) − P ( 15 ) = 51 .
The rate of population change 15 months from now is 50 people per month, and the actual change 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 the rate of change of the population 15 months from now and the actual change in population during the 16th month.
Finding the Derivative (a) To find the rate of change of the population with respect to time, we need to find the derivative of the population function P ( x ) with respect to 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
Evaluating the Derivative at x=15 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
Calculating Population at x=15 and x=16 (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 the population at the end of the 16th month, P ( 16 ) . Then, we find the difference between P ( 16 ) and P ( 15 ) .
P ( 15 ) = ( 15 ) 2 + 20 ( 15 ) + 8000 = 225 + 300 + 8000 = 8525
P ( 16 ) = ( 16 ) 2 + 20 ( 16 ) + 8000 = 256 + 320 + 8000 = 8576
Finding the Change in Population The actual change in population during the 16th month is:
P ( 16 ) − P ( 15 ) = 8576 − 8525 = 51
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 in urban planning. For instance, predicting how quickly a city's population will grow helps in planning infrastructure like schools, hospitals, and transportation. By using functions and derivatives, city planners can estimate these changes and make informed decisions about resource allocation to meet the needs of the growing community. This ensures sustainable development and improves the quality of life for residents.
