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Suppose that
$12,000 is invested in a bond fund and the account grows to
$14,309.26 in 4 yrs.
a. Use the model
A=Pe^(rt) to determine the average rate of return under continuous compounding. Round to the nearest tenth of a percent.

Suppose that $\$ 12,000$ is invested in a bond fund and the account grows to $\$ 14,309.26$ in $4$ yrs. $\newline$ a. Use the model $A=P e^{r t}$ to determine the average rate of return under continuous compounding. Round to the nearest tenth of a percent.

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Volume of cubes and rectangular prisms: word problems

Full solution

Q. Suppose that

\$ 12,000

is invested in a bond fund and the account grows to

\$ 14,309.26

in

4

yrs.

\newline

a. Use the model

A=P e^{r t}

to determine the average rate of return under continuous compounding. Round to the nearest tenth of a percent.

Formula Explanation: We have the formula $A=Pe^{rt}$ where $A$ is the amount after time $t$ , $P$ is the principal amount, $r$ is the rate of return, and $e$ is the base of the natural logarithm.
Given Values: We know $A=(\$)14,309.26$ , $P=(\$)12,000$ , and $t=4$ years. We need to find $r$ .
Isolate r: First, let's isolate r in the equation. We divide both sides by P to get $\frac{A}{P} = e^{rt}$ .
Plug in Numbers: Now we plug in the numbers: $14309.26/12000 = e^{4r}$ .
Calculate Left Side: Calculate the left side: $14309.26/12000 = 1.192438333$ .
Take Natural Logarithm: Next, we take the natural logarithm of both sides to get $\ln(1.192438333) = \ln(e^{4r})$ which simplifies to $\ln(1.192438333) = 4r$ .
Calculate Natural Log: Calculate $\ln(1.192438333)$ to get $r$ . $\ln(1.192438333) = 0.177627958$ .
Divide by $4$ : Now, divide by $4$ to find $r$ : $0.177627958/4 = 0.0444069895$ .
Express as Percentage: To express $r$ as a percentage, we multiply by $100$ : $0.0444069895 \times 100 = 4.44069895\%$ .
Round to Nearest Tenth: Round to the nearest tenth of a percent: $r \approx 4.4\%$ .

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