  ### Home > CALC > Chapter 8 > Lesson 8.3.1 > Problem8-105

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?

1. 18,351

2. 20,250

3. 21,348

4. 22,571

5. 23,374

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?

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

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

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

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

Integrate both sides.

Combine the constants of integration.

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

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

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