CPM Homework Help : CALC Problem 8-105

Multiple Choice: The population $P$ of a city is growing according to the equation $\frac { d P } { d t }= 0.02P + 357$ . The current population of the city is $18{,}000$ . Assuming the same pattern of growth, what will be the population be in six years?

$18{,}351$

$20{,}250$

$21{,}348$

$22{,}571$

$23{,}374$

Hint 1:

Notice that the given equation is a derivative. It gives information about the rate of change of the population. But what we want to know is information about the actual population. How do we undo a derivative?

Hint 2:

It is necessary to use implicit integration because the derivative is in terms of time, but time does not appear in the equation.

Step 1:

$\frac{dP}{dt}=0.02P+357$
$\frac{dP}{dt}=0.02(P+17850)$

Step 2:

Separate the sides: $\frac{1}{P+17850}dP=0.02dt$

Step 3:

Integrate both sides.

Step 4:

Combine the constants of integration.

Step 5:

Evaluate the given point to solve for $C$ : $C = 35800$

Step 6:

Substitute in the value of $C$ , then write an equation for $P(t)$ .

Step 7:

Use your $P(t)$ equation to find Population at $t = 6$ years.