Question

if an amount $p_0$ 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 $\frac{dp}{dt} = 0.071p$.
a) find the function that satisfies the equation. write it in terms of $p_0$ and 0.071.
b) suppose that $1500 is invested. what is the balance after 1 year? after 4 years?
c) what is the rate of change of the balance after 1 year? after 4 years?
a) $p(t) = \square$
b) the balance after 1 year is $\square$ the balance after 4 years is $\square$
(type integers or decimals rounded to two decimal places as needed.)
c) the rate of change of the balance after 1 year is $\square$ $\boldsymbol{\
abla}$ the rate of change of the balance after 4 years is $\square$ $\boldsymbol{\
abla}$
(type integers or decimals rounded to two decimal places as needed.)

Explanation:

Step1: Solve the differential equation

The equation $\frac{dP}{dt}=0.071P$ is separable. Separate variables: $\frac{dP}{P}=0.071dt$. Integrate both sides: $\ln|P|=0.071t+C$. Exponentiate: $P(t)=P_0e^{0.071t}$ (where $P_0=P(0)$).

Step2: Calculate balance after 1 year

Substitute $P_0=1500$, $t=1$: $P(1)=1500e^{0.071×1}≈1500×1.0736≈1610.40$.

Step3: Calculate balance after 4 years

Substitute $P_0=1500$, $t=4$: $P(4)=1500e^{0.071×4}=1500e^{0.284}≈1500×1.328≈1992.00$.

Step4: Find rate of change at t=1

Rate is $\frac{dP}{dt}=0.071P(t)$. Use $P(1)≈1610.40$: $0.071×1610.40≈114.34$.

Step5: Find rate of change at t=4

Use $P(4)≈1992.00$: $0.071×1992.00≈141.43$.

Answer:

a) $P(t)=P_0e^{0.071t}$
b) 1610.40; 1992.00
c) 114.34; 141.43