  ### Home > APCALC > Chapter 8 > Lesson 8.3.1 > Problem8-106

8-106.

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. $18,351$

1. $20,250$

1. $21,348$

1. $22,571$

1. $23,374$

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 $C$: $C = 35850$

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.