if an amount p0 is invested in the mandelbrot bond fund and interest is compounded continuously at 7.1% per…

if an amount p0 is invested in the mandelbrot bond fund and interest is compounded continuously at 7.1% per year, the balance p grows at the rate given by dp/dt = 0.071p. a) find the function that satisfies the equation. write it in terms of p0 and 0.071. b) suppose that $1500 is invested. what is the balance after 1 year? after 3 years? c) what is the rate of change of the balance after 1 year? after 3 years? a) p(t)=p0e^0.071t b) the balance after 1 year is $ . the balance after 3 years is $ . (type integers or decimals rounded to two decimal places as needed.)

if an amount p0 is invested in the mandelbrot bond fund and interest is compounded continuously at 7.1% per year, the balance p grows at the rate given by dp/dt = 0.071p. a) find the function that satisfies the equation. write it in terms of p0 and 0.071. b) suppose that $1500 is invested. what is the balance after 1 year? after 3 years? c) what is the rate of change of the balance after 1 year? after 3 years? a) p(t)=p0e^0.071t b) the balance after 1 year is $ . the balance after 3 years is $ . (type integers or decimals rounded to two decimal places as needed.)

Answer

Explanation:

Step1: Substitute values for part b

For $t = 1$ and $P_0=1500$ in $P(t)=P_0e^{0.071t}$, we have $P(1)=1500e^{0.071\times1}$.

Step2: Calculate $P(1)$

$P(1)=1500e^{0.071}\approx1500\times1.0735 = 1610.25$.

Step3: Substitute values for $t = 3$

For $t = 3$ and $P_0 = 1500$ in $P(t)=P_0e^{0.071t}$, we have $P(3)=1500e^{0.071\times3}$.

Step4: Calculate $P(3)$

$P(3)=1500e^{0.213}\approx1500\times1.2377=1856.55$.

Step5: Find derivative for part c

The derivative of $P(t)=P_0e^{0.071t}$ with respect to $t$ is $\frac{dP}{dt}=0.071P_0e^{0.071t}$.

Step6: Calculate rate of change at $t = 1$

When $t = 1$ and $P_0 = 1500$, $\frac{dP}{dt}\big|_{t = 1}=0.071\times1500e^{0.071}\approx0.071\times1610.25\approx114.33$.

Step7: Calculate rate of change at $t = 3$

When $t = 3$ and $P_0 = 1500$, $\frac{dP}{dt}\big|_{t = 3}=0.071\times1500e^{0.213}\approx0.071\times1856.55\approx131.81$.

Answer:

b) The balance after 1 year is $$1610.25$. The balance after 3 years is $$1856.55$. c) The rate of change of the balance after 1 year is $$114.33$. The rate of change of the balance after 3 years is $$131.81$.