Math / Calculus

Question20,100 is invester at $3.50^{\circ}$ intrest compounded continously. how lony will it tuke for the investment to double.

Studdy Solution

STEP 1

What is this asking? How long does it take for an initial investment of $\$20,100$ to double, given a continuously compounded interest rate of $3.5\%$ ? Watch out! Remember to convert the interest rate from a percentage to a decimal and be careful with the continuous compounding formula!

STEP 2

1. Define the formula
2. Set up the equation
3. Solve for time

STEP 3

We'll use the formula for continuous compound interest: $A = P \cdot e^{rt}$ Where: $A$ is the final amount, $P$ is the principal (initial investment), $r$ is the interest rate (as a decimal), $t$ is the time (in years), and $e$ is the mathematical constant approximately equal to $2.71828$ .

STEP 4

This formula tells us how much money we'll have ( $A$ ) after a certain amount of time ( $t$ ) given an initial investment ( $P$ ) and a continuously compounded interest rate ( $r$ ).

STEP 5

We know: $P = \$20,100$ (initial investment) $r = 0.035$ (interest rate of $3.5\%$ expressed as a decimal) We want to find the time ( $t$ ) when the investment doubles, so $A = 2P = 2 \cdot \$20,100 = \$40,200$ .

STEP 6

Now, let's plug these values into our formula: $\$40,200 = \$20,100 \cdot e^{0.035t}$

STEP 7

To solve for $t$ , we first need to isolate the exponential term.
We can do this by dividing both sides of the equation by the principal amount, $\$20,100$ : $\frac{\$40,200}{\$20,100} = \frac{\$20,100 \cdot e^{0.035t}}{\$20,100}$ $2 = e^{0.035t}$

STEP 8

Next, we'll apply the natural logarithm (ln) to both sides.
The natural logarithm is the inverse of the exponential function with base $e$ , so it helps us "undo" the exponential: $\ln(2) = \ln(e^{0.035t})$ $\ln(2) = 0.035t$

STEP 9

Finally, we can isolate $t$ by dividing both sides by $0.035$ : $t = \frac{\ln(2)}{0.035}$ $t \approx \frac{0.6931}{0.035}$ $t \approx 19.80 \text{ years}$

STEP 10

It will take approximately 19.80 years for the investment to double.

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