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 formula for continuous compounding is given by $A(t) = A_0 e^{rt}$, where $A_0$ is the initial amount, $r$ is the interest rate, and $t$ is time. Taking the derivative with respect to $t$, we get $\frac{dA}{dt} = rA$. Since the interest rate is $20\%$, we have $r=0.20$, so the differential equation is $\frac{dA}{dt} = 0.20 A$. The initial condition is $A(0) = 2,000$, since that is the initial amount invested. Therefore, the correct choice is 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 $A ( t )$