suppose that $p_0$ is invested in a savings account in which interest is compounded continuously at 6.6% per…

suppose that $p_0$ is invested in a savings account in which interest is compounded continuously at 6.6% per year. that is, the balance $p$ grows at the rate given by the following equation $\frac{dp}{dt}=0.066p(t)$ (a)find the function $p(t)$ that satisfies the equation. write it in terms of $p_0$ and 0.066. (b)suppose that $1000 is invested. what is the balance after 3 years? (c)when will an investment of $1000 double itself? (a) choose the correct answer below a. $p(t)=0.066p_0e^t$ b. $p(t)=p_0e^{0.066t}$ c. $p(t)=p(t)e^{0.066t}$ d. $p_0 = p(t)e^{0.066t}$

suppose that $p_0$ is invested in a savings account in which interest is compounded continuously at 6.6% per year. that is, the balance $p$ grows at the rate given by the following equation $\frac{dp}{dt}=0.066p(t)$ (a)find the function $p(t)$ that satisfies the equation. write it in terms of $p_0$ and 0.066. (b)suppose that $1000 is invested. what is the balance after 3 years? (c)when will an investment of $1000 double itself? (a) choose the correct answer below a. $p(t)=0.066p_0e^t$ b. $p(t)=p_0e^{0.066t}$ c. $p(t)=p(t)e^{0.066t}$ d. $p_0 = p(t)e^{0.066t}$

Answer

Explanation:

Step1: Solve the differential equation

The differential equation $\frac{dP}{dt}=0.066P$ is a first - order separable differential equation. We can rewrite it as $\frac{dP}{P}=0.066dt$. Integrating both sides: $\int\frac{dP}{P}=\int0.066dt$. The left - hand side integral is $\ln|P|$ and the right - hand side is $0.066t + C$. So, $\ln|P| = 0.066t+C$. Exponentiating both sides gives $P = Ce^{0.066t}$. When $t = 0$, $P(0)=P_0$, so $P_0 = C$. Thus, $P(t)=P_0e^{0.066t}$.

Step2: Find the balance after 3 years

If $P_0 = 1000$ and $t = 3$, then $P(3)=1000e^{0.066\times3}=1000e^{0.198}\approx1000\times1.2197=1219.7$.

Step3: Find the time for the investment to double

If the investment doubles, then $P(t)=2P_0$. Substituting into $P(t)=P_0e^{0.066t}$, we get $2P_0=P_0e^{0.066t}$. Divide both sides by $P_0$ (since $P_0\neq0$), we have $2 = e^{0.066t}$. Take the natural logarithm of both sides: $\ln(2)=\ln(e^{0.066t})$. Since $\ln(e^{x})=x$, then $\ln(2)=0.066t$. Solving for $t$, we get $t=\frac{\ln(2)}{0.066}\approx\frac{0.6931}{0.066}\approx10.5$.

Answer:

(a) B. $P(t)=P_0e^{0.066t}$ (b) $$1219.7$ (c) Approximately $10.5$ years