Complete the table for a savings account in which interest is compounded continuously.
initial investment = $600
annual % rate = ?
time to double ?
Amount after 10 years = $19,205.00
I have no idea how to solve this.
Neither do I, since you don't have enough information.
Either you have to tell me what the rate is, or the time for doubling.
sorry, you do have enough information, but please please tell me where I can invest $600 and have it grow to $19205 in 10 years
19205 = 600(1+i)^10
32.008 = (1+i)^10
take the 10th root
1.414 = 1+i
i = .414 or 41.4% interest if compounded annually
To solve this problem, we need to use the continuous compound interest formula:
A = P * e^(rt)
Where:
A = the final amount
P = the initial investment
e = Euler's number (approximately 2.71828)
r = the annual interest rate (in decimal form)
t = the time in years
Let's solve the problem step-by-step:
1. We are given the initial investment, P = $600.
2. We are also given the final amount, A = $19,205.00 after 10 years.
To find the annual interest rate, we need to solve for r. Since we know the initial investment and the final amount, we can rearrange the formula as follows:
A = P * e^(rt)
19,205 = 600 * e^(r * 10)
3. Divide both sides of the equation by 600:
19,205 / 600 = e^(r * 10)
4. Take the natural logarithm (ln) of both sides to isolate r * 10:
ln(19,205 / 600) = r * 10
5. Divide both sides of the equation by 10:
ln(19,205 / 600) / 10 = r
Using a calculator, calculate ln(19,205 / 600) / 10 to find the value of r.
6. The calculated value of r will be the annual interest rate in decimal form.
Now let's calculate the time it takes to double the initial investment:
To find the time it takes to double the initial investment, we can use the rule of 72. The rule of 72 states that if you divide 72 by the annual interest rate (in percent form), the result will give you the approximate number of years it takes for the investment to double.
Let's use this rule to find the time to double:
72 / (annual interest rate in percent form) = time to double
Using the calculated annual interest rate from step 6, divide 72 by the annual interest rate to find the time it takes to double.
I hope this helps!
To find the missing information for a savings account with continuous compounding, we can use the formula for the amount of money accumulated over time:
A = P * e^(rt)
Where:
A is the amount of money accumulated
P is the initial investment
e is the mathematical constant approximately equal to 2.71828
r is the annual interest rate (in decimal form)
t is the time in years
Let's start by finding the annual interest rate, r.
1. Plug in the given information to the formula:
A = $19,205.00
P = $600
t = 10 years
19,205.00 = 600 * e^(10r)
2. Divide both sides by 600:
32.01 = e^(10r)
3. Take the natural logarithm (ln) of both sides to isolate the exponential term:
ln(32.01) = 10r
4. Divide both sides by 10, giving you the value of r:
r ≈ ln(32.01) / 10
Using a calculator, we find that r ≈ 0.219838.
This means the annual interest rate is approximately 0.219838, or about 21.9838%.
Now let's find the time it takes to double the initial investment.
1. Use the formula for doubling time:
A = P * e^(rt)
2. Set A equal to 2P since we want to find the time when the amount doubles:
2P = P * e^(rt)
3. Divide both sides by P:
2 = e^(rt)
4. Take the natural logarithm (ln) of both sides:
ln(2) = rt
5. Divide both sides by r to solve for t:
t = ln(2) / r
Using the value of r from before (approximately 0.219838), we can calculate the time it takes to double the initial investment:
t ≈ ln(2) / 0.219838
Using a calculator, we find that t ≈ 3.1538 years.
Therefore, the time it takes for the initial investment to double is approximately 3.1538 years.