use the present - value formula to determine the amount to be invested now, or the present value needed. the…

use the present - value formula to determine the amount to be invested now, or the present value needed. the desired accumulated amount is $40,000 after 14 years invested in an account with 4.7% interest compounded monthly. the amount to be invested now, or the present value needed, is $ (round to the nearest cent as needed.)

use the present - value formula to determine the amount to be invested now, or the present value needed. the desired accumulated amount is $40,000 after 14 years invested in an account with 4.7% interest compounded monthly. the amount to be invested now, or the present value needed, is $ (round to the nearest cent as needed.)

Answer

Explanation:

Step1: Identify the compound - interest formula for present value

The compound - interest formula for present value is $PV=\frac{FV}{(1 + \frac{r}{n})^{nt}}$, where $PV$ is the present value (the amount to be invested now), $FV$ is the future value (the desired accumulated amount), $r$ is the annual interest rate (in decimal form), $n$ is the number of times interest is compounded per year, and $t$ is the number of years.

Step2: Convert the given values to the appropriate form

We are given that $FV = 40000$, $r=0.047$ (since $4.7%=0.047$), $n = 12$ (compounded monthly), and $t = 14$.

Step3: Substitute the values into the formula

$PV=\frac{40000}{(1+\frac{0.047}{12})^{12\times14}}$. First, calculate the value inside the parentheses: $\frac{0.047}{12}\approx0.0039167$, then $1+\frac{0.047}{12}\approx1.0039167$. Next, calculate the exponent: $12\times14 = 168$. So, $(1+\frac{0.047}{12})^{12\times14}=(1.0039167)^{168}$. Using a calculator, $(1.0039167)^{168}\approx1.90799$. Then, $PV=\frac{40000}{1.90799}\approx20964.58$.

Answer:

$20964.58$