Suppose that $\$ 12,000$ is invested in a bond fund and the account grows to $\$ 14,309.26$ in $4$ yrs. $\newline$ b. How long will it take the investment to reach $\$ 20,000$ if the rate of return continues? Round to the nearest tenth of a year.

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Q. Suppose that

\$ 12,000

is invested in a bond fund and the account grows to

\$ 14,309.26

4

yrs.

\newline

b. How long will it take the investment to reach

\$ 20,000

if the rate of return continues? Round to the nearest tenth of a year.

Calculate Rate of Return: First, calculate the rate of return using the initial and final amounts over the $4$ -year period. Use the formula for compound interest: $A = P(1 + r/n)^{(nt)}$ , where $A$ is the amount of money accumulated after $n$ years, including interest, $P$ is the principal amount, $r$ is the annual interest rate, $n$ is the number of times that interest is compounded per year, and $t$ is the time the money is invested for. Since we don't know $n$ and $r$ separately, we'll assume the interest is compounded once per year, so $A = P(1 + r/n)^{(nt)}$ $0$ . Then we solve for $r$ .
Simplify Equation: Plug in the known values: $\$14,309.26 = \$12,000(1 + \frac{r}{1})^{1\times4}$ .
Isolate $(1 + r)^4$ : Simplify the equation: $\$14,309.26 = \$12,000(1 + r)^4$ .
Calculate Fourth Root: Divide both sides by $\$12,000$ to isolate $(1 + r)^4$ : $(1 + r)^4 = \frac{\$14,309.26}{\$12,000}$ .
Find $r$ : Calculate the right side: $(1 + r)^4 = 1.192438333$ .
Use Rate of Return: Take the fourth root of both sides to solve for $(1 + r)$ : $1 + r = (1.192438333)^{1/4}$ .
Isolate $(1 + 0.045)^t$ : Calculate the fourth root: $1 + r = 1.045$ .
Calculate Natural Logarithm: Subtract $1$ from both sides to find $r$ : $r = 1.045 - 1$ .
Solve for $t$ : Calculate $r$ : $r = 0.045$ or $4.5\%$ .
Round to Nearest Tenth: Now, use the rate of return to find out how long it will take for the investment to grow to $\$20,000$ . Use the same compound interest formula: $\$20,000 = \$12,000(1 + 0.045)^{(1\cdot t)}$ .
Round to Nearest Tenth: Now, use the rate of return to find out how long it will take for the investment to grow to $\$20,000$ . Use the same compound interest formula: $\$20,000 = \$12,000(1 + 0.045)^{1\cdot t}$ .Divide both sides by $\$12,000$ to isolate $(1 + 0.045)^t$ : $(1 + 0.045)^t = \frac{\$20,000}{\$12,000}$ .
Round to Nearest Tenth: Now, use the rate of return to find out how long it will take for the investment to grow to $\$20,000$ . Use the same compound interest formula: $\$20,000 = \$12,000(1 + 0.045)^{1\cdot t}$ .Divide both sides by $\$12,000$ to isolate $(1 + 0.045)^t$ : $(1 + 0.045)^t = \frac{\$20,000}{\$12,000}$ .Calculate the right side: $(1 + 0.045)^t = 1.666666667$ .
Round to Nearest Tenth: Now, use the rate of return to find out how long it will take for the investment to grow to $\$20,000$ . Use the same compound interest formula: $\$20,000 = \$12,000(1 + 0.045)^{1\cdot t}$ .Divide both sides by $\$12,000$ to isolate $(1 + 0.045)^t$ : $(1 + 0.045)^t = \frac{\$20,000}{\$12,000}$ .Calculate the right side: $(1 + 0.045)^t = 1.666666667$ .Take the natural logarithm of both sides to solve for $t$ : $\ln((1 + 0.045)^t) = \ln(1.666666667)$ .
Round to Nearest Tenth: Now, use the rate of return to find out how long it will take for the investment to grow to $\$20,000$ . Use the same compound interest formula: $\$20,000 = \$12,000(1 + 0.045)^{1\cdot t}$ .Divide both sides by $\$12,000$ to isolate $(1 + 0.045)^t$ : $(1 + 0.045)^t = \frac{\$20,000}{\$12,000}$ .Calculate the right side: $(1 + 0.045)^t = 1.666666667$ .Take the natural logarithm of both sides to solve for $t$ : $\ln((1 + 0.045)^t) = \ln(1.666666667)$ .Use the property of logarithms that $\ln(a^b) = b\cdot\ln(a)$ : $t\cdot\ln(1 + 0.045) = \ln(1.666666667)$ .
Round to Nearest Tenth: Now, use the rate of return to find out how long it will take for the investment to grow to $\$20,000$ . Use the same compound interest formula: $\$20,000 = \$12,000(1 + 0.045)^{(1*t)}$ .Divide both sides by $\$12,000$ to isolate $(1 + 0.045)^t$ : $(1 + 0.045)^t = \frac{\$20,000}{\$12,000}$ .Calculate the right side: $(1 + 0.045)^t = 1.666666667$ .Take the natural logarithm of both sides to solve for $t$ : $\ln((1 + 0.045)^t) = \ln(1.666666667)$ .Use the property of logarithms that $\ln(a^b) = b*\ln(a)$ : $t*\ln(1 + 0.045) = \ln(1.666666667)$ .Calculate the natural logarithm of $\$20,000 = \$12,000(1 + 0.045)^{(1*t)}$ $0$ and $\$20,000 = \$12,000(1 + 0.045)^{(1*t)}$ $1$ : $\$20,000 = \$12,000(1 + 0.045)^{(1*t)}$ $2$ .
Round to Nearest Tenth: Now, use the rate of return to find out how long it will take for the investment to grow to $\$20,000$ . Use the same compound interest formula: $\$20,000 = \$12,000(1 + 0.045)^{1\cdot t}$ .Divide both sides by $\$12,000$ to isolate $(1 + 0.045)^t$ : $(1 + 0.045)^t = \frac{\$20,000}{\$12,000}$ .Calculate the right side: $(1 + 0.045)^t = 1.666666667$ .Take the natural logarithm of both sides to solve for $t$ : $\ln((1 + 0.045)^t) = \ln(1.666666667)$ .Use the property of logarithms that $\ln(a^b) = b\cdot\ln(a)$ : $t\cdot\ln(1 + 0.045) = \ln(1.666666667)$ .Calculate the natural logarithm of $\$20,000 = \$12,000(1 + 0.045)^{1\cdot t}$ $0$ and $\$20,000 = \$12,000(1 + 0.045)^{1\cdot t}$ $1$ : $\$20,000 = \$12,000(1 + 0.045)^{1\cdot t}$ $2$ .Calculate the values: $\$20,000 = \$12,000(1 + 0.045)^{1\cdot t}$ $3$ .
Round to Nearest Tenth: Now, use the rate of return to find out how long it will take for the investment to grow to $\$20,000$ . Use the same compound interest formula: $\$20,000 = \$12,000(1 + 0.045)^{1\cdot t}$ .Divide both sides by $\$12,000$ to isolate $(1 + 0.045)^t$ : $(1 + 0.045)^t = \frac{\$20,000}{\$12,000}$ .Calculate the right side: $(1 + 0.045)^t = 1.666666667$ .Take the natural logarithm of both sides to solve for $t$ : $\ln((1 + 0.045)^t) = \ln(1.666666667)$ .Use the property of logarithms that $\ln(a^b) = b\cdot\ln(a)$ : $t\cdot\ln(1 + 0.045) = \ln(1.666666667)$ .Calculate the natural logarithm of $\$20,000 = \$12,000(1 + 0.045)^{1\cdot t}$ $0$ and $\$20,000 = \$12,000(1 + 0.045)^{1\cdot t}$ $1$ : $\$20,000 = \$12,000(1 + 0.045)^{1\cdot t}$ $2$ .Calculate the values: $\$20,000 = \$12,000(1 + 0.045)^{1\cdot t}$ $3$ .Divide both sides by $\$20,000 = \$12,000(1 + 0.045)^{1\cdot t}$ $4$ to solve for $t$ : $\$20,000 = \$12,000(1 + 0.045)^{1\cdot t}$ $6$ .
Round to Nearest Tenth: Now, use the rate of return to find out how long it will take for the investment to grow to $\$20,000$ . Use the same compound interest formula: $\$20,000 = \$12,000(1 + 0.045)^{1\cdot t}$ .Divide both sides by $\$12,000$ to isolate $(1 + 0.045)^t$ : $(1 + 0.045)^t = \frac{\$20,000}{\$12,000}$ .Calculate the right side: $(1 + 0.045)^t = 1.666666667$ .Take the natural logarithm of both sides to solve for $t$ : $\ln((1 + 0.045)^t) = \ln(1.666666667)$ .Use the property of logarithms that $\ln(a^b) = b\cdot\ln(a)$ : $t\cdot\ln(1 + 0.045) = \ln(1.666666667)$ .Calculate the natural logarithm of $\$20,000 = \$12,000(1 + 0.045)^{1\cdot t}$ $0$ and $\$20,000 = \$12,000(1 + 0.045)^{1\cdot t}$ $1$ : $\$20,000 = \$12,000(1 + 0.045)^{1\cdot t}$ $2$ .Calculate the values: $\$20,000 = \$12,000(1 + 0.045)^{1\cdot t}$ $3$ .Divide both sides by $\$20,000 = \$12,000(1 + 0.045)^{1\cdot t}$ $4$ to solve for $t$ : $\$20,000 = \$12,000(1 + 0.045)^{1\cdot t}$ $6$ .Calculate $t$ : $\$20,000 = \$12,000(1 + 0.045)^{1\cdot t}$ $8$ years.
Round to Nearest Tenth: Now, use the rate of return to find out how long it will take for the investment to grow to $\$20,000$ . Use the same compound interest formula: $\$20,000 = \$12,000(1 + 0.045)^{1\cdot t}$ .Divide both sides by $\$12,000$ to isolate $(1 + 0.045)^t$ : $(1 + 0.045)^t = \frac{\$20,000}{\$12,000}$ .Calculate the right side: $(1 + 0.045)^t = 1.666666667$ .Take the natural logarithm of both sides to solve for $t$ : $\ln((1 + 0.045)^t) = \ln(1.666666667)$ .Use the property of logarithms that $\ln(a^b) = b\cdot\ln(a)$ : $t\cdot\ln(1 + 0.045) = \ln(1.666666667)$ .Calculate the natural logarithm of $\$20,000 = \$12,000(1 + 0.045)^{1\cdot t}$ $0$ and $\$20,000 = \$12,000(1 + 0.045)^{1\cdot t}$ $1$ : $\$20,000 = \$12,000(1 + 0.045)^{1\cdot t}$ $2$ .Calculate the values: $\$20,000 = \$12,000(1 + 0.045)^{1\cdot t}$ $3$ .Divide both sides by $\$20,000 = \$12,000(1 + 0.045)^{1\cdot t}$ $4$ to solve for $t$ : $\$20,000 = \$12,000(1 + 0.045)^{1\cdot t}$ $6$ .Calculate $t$ : $\$20,000 = \$12,000(1 + 0.045)^{1\cdot t}$ $8$ years.Round to the nearest tenth of a year: $\$20,000 = \$12,000(1 + 0.045)^{1\cdot t}$ $9$ years.