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Suppose that at 19 years old you win
$100,000 playing the lottery. If you would like to have
$1,000,000 when you retire at age 67 , determine the average rate of return needed under continuous compounding.

$3$ . Suppose that at $19$ years old you win $\$ 100,000$ playing the lottery. If you would like to have $\$ 1,000,000$ when you retire at age $67$ , determine the average rate of return needed under continuous compounding.

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Linear Inequality Word Problems

Full solution

Q.

3

. Suppose that at

19

years old you win

\$ 100,000

playing the lottery. If you would like to have

\$ 1,000,000

when you retire at age

67

, determine the average rate of return needed under continuous compounding.

Continuous Compounding Formula: We use the formula for continuous compounding: $A = Pe^{rt}$ , where $A$ is the amount of money accumulated after $n$ years, including interest, $P$ is the principal amount, $r$ is the rate of interest per year, and $t$ is the time in years.
Plug in Values: First, let's plug in the values we know. We want $A$ to be $\$1,000,000$ , $P$ is $\$100,000$ , and $t$ is the number of years from $19$ to $67$ , which is $67 - 19 = 48$ years.
Isolate Variable: Now we have $\$1,000,000 = \$100,000 \times e^{48r}$ . To solve for $r$ , we need to isolate it on one side of the equation.
Take Natural Logarithm: Divide both sides by $\$100,000$ to get $10 = e^{48r}$ .
Apply Logarithm Property: Take the natural logarithm ( $\ln$ ) of both sides to get $\ln(10) = \ln(e^{48r})$ .
Solve for $r$ : Using the property of logarithms that $\ln(e^x) = x$ , we have $\ln(10) = 48r$ .
Calculate Final Value: Divide both sides by $48$ to solve for $r$ : $r = \frac{\ln(10)}{48}$ .
Calculate Final Value: Divide both sides by $48$ to solve for $r$ : $r = \frac{\ln(10)}{48}$ .Calculate $r$ using a calculator: $r = \frac{\ln(10)}{48} \approx 0.05776226505$ or about $5.78\%$ .

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