A sum of $\$ 2,000$ is invested at $20 \%$ interest. If $A ( t )$ is the amount of the investment at time $t$ for the case of continuous compounding, write a differential equation and an initial condition satisfied by $A ( t )$.
A) $\frac { d A } { d t } = 20 A , \quad A ( 0 ) = 3,000$
B) $\frac { d A } { d t } = 1,500 A , \quad A ( 0 ) = 2,000$
C) $\frac { d A } { d t } = 0.20 A , A ( 0 ) = 200$
D) $\frac { d A } { d t } = 0.20 A ( 0 ) , \quad A ( 0 ) = 2,000$
E) $\frac { d A } { d t } = 0.20 A , \quad A ( 0 ) = 2,000$

Question

Quizplus · Accepted Answer

The differential equation for continuous compounding is given by $\frac{dA}{dt}=rA$, where $r$ is the interest rate. In this case, $r=20\%=0.20$. Therefore, the differential equation is $\frac{dA}{dt}=0.20A$. The initial condition is $A(0)=2,000$, since that is the initial investment amount. Thus, the correct choice is $\boxed{	extbf{(E)}}$.

A Sum Of $2,000\$ 2,000$2,000 Is Invested At 20%20 \%20% Interest If A(t)A ( t )A(t)

A Sum Of $\$ 2,000$ Is Invested At $20 \%$ Interest If $A ( t )$