Question

if an amount $p_0$ is invested in the mandelbrot bond fund and interest is compounded continuously at 5.8% per year, the balance $p$ grows at the rate given by $\frac{dp}{dt}=0.058p$. a) find the function that satisfies the equation. write it in terms of $p_0$ and 0.058. 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) $p(t)=p_0e^{0.058t}$ b) the balance after 1 year is $ $. the balance after 3 years is $ $ (type integers or decimals rounded to two decimal places as needed.)

Explanation:

Step1: Calculate balance after 1 year

Given $P_0 = 1500$, $t = 1$, and the formula $P(t)=P_0e^{0.058t}$. Substitute the values into the formula: $P(1)=1500e^{0.058\times1}=1500e^{0.058}$. Using a calculator, $P(1)\approx1500\times1.05973 = 1589.60$.

Step2: Calculate balance after 3 years

Substitute $P_0 = 1500$, $t = 3$ into the formula $P(t)=P_0e^{0.058t}$. So $P(3)=1500e^{0.058\times3}=1500e^{0.174}$. Using a calculator, $P(3)\approx1500\times1.19034=1785.51$.

Answer:

The balance after 1 year is $\$1589.60$.
The balance after 3 years is $\$1785.51$.