find the principal needed now to get the given amount; that is, find the present value. to get $4000 after…

Explanation:

Step1: Identify the compound - interest formula for present value

The compound - interest formula for present value $P$ when compounded $n$ times a year is $P=\frac{A}{(1 + \frac{r}{n})^{nt}}$, where $A$ is the future value, $r$ is the annual interest rate (in decimal form), $n$ is the number of times compounded per year, and $t$ is the number of years. Given $A = 4000$, $r=0.11$ (since $11%=0.11$), $n = 365$ (compounded daily), and $t = 2.5$ years.

Step2: Substitute the values into the formula

$P=\frac{4000}{(1+\frac{0.11}{365})^{365\times2.5}}$. First, calculate the exponent: $365\times2.5 = 912.5$. Then, calculate the value inside the parentheses: $\frac{0.11}{365}\approx0.00030137$, and $1+\frac{0.11}{365}=1 + 0.00030137=1.00030137$. Next, calculate $(1.00030137)^{912.5}$. Using a calculator, $(1.00030137)^{912.5}\approx1.31677$. Finally, $P=\frac{4000}{1.31677}\approx3037.6$.

Answer:

$3037.60$