Complete the table for a savings account in which interest is compounded continuously. (Round your answers to four decimal places.)
Initial Investment: $350
Annual % Rate: %
Time to Double: yr
Amount After 10 Years: $435.21
350 e^(10r) = 435.21
e^(10r) = 1.243457...
take ln of both sides:
10r = ln(1.243457..) = .2178955..
r = .02178955... or appr 2.1790%
now that you know the rate, for the doubling time ....
350e^(.02178955..t) = 700
I will let you solve for t
girl i don’t get this
To complete the table for a savings account with continuous compounding, we need to find the annual interest rate and the time it takes to double the initial investment.
To find the annual interest rate, we can use the formula for continuous compounding:
A = P * e^(rt),
where A is the final amount, P is the initial investment, e is Euler's number (approximately 2.71828), r is the annual interest rate, and t is the time in years.
Let's solve for 'r' first:
A = P * e^(rt)
435.21 = 350 * e^(r * 10)
Divide both sides by 350:
435.21 / 350 = e^(10r)
Take the natural logarithm (ln) of both sides:
ln(435.21 / 350) = ln(e^(10r))
By the properties of logarithms, we can bring down the exponent:
ln(435.21 / 350) = 10r * ln(e)
Since ln(e) equals 1, the equation simplifies to:
ln(435.21 / 350) = 10r
Now, we can solve for 'r' using a calculator. Let's assume we find that r is equal to 0.04 (4%).
Next, to find the time it takes to double the initial investment, we can use the formula:
t = ln(2) / r
Let's calculate the time:
t = ln(2) / 0.04
Using a calculator, we find t is equal to approximately 17.33 years.
Finally, to find the amount after 10 years, we can use the formula:
A = P * e^(rt)
Let's calculate the amount:
A = 350 * e^(0.04 * 10)
Using a calculator, we find that A is equal to approximately $483.49.
Therefore, the completed table is as follows:
Initial Investment: $350
Annual % Rate: 4% (approximately)
Time to Double: 17.33 years (approximately)
Amount After 10 Years: $483.49 (approximately)
To complete the table for a savings account with continuous compounding, we need to find the annual interest rate and the time it takes for the investment to double.
Let's start with finding the annual interest rate. To do that, we will use the formula for continuous compounding:
A = P * e^(rt)
Where:
A = the final amount of the investment
P = the initial investment
e = Euler's number (approximately 2.71828)
r = the annual interest rate (in decimal form)
t = time in years
We can rearrange the equation to solve for r:
r = ln(A / P) / t
Given that the initial investment (P) is $350 and the amount after 10 years (A) is $435.21, we can plug in the values into the equation:
r = ln(435.21 / 350) / 10
Using a scientific calculator or an online calculator, we can find that r is approximately 0.0241.
Now that we have the annual interest rate, we can find the time it takes for the investment to double. The rule of 72 is often used to estimate the time it takes for an investment to double. It's calculated as:
Time to double = 72 / r
Plugging in the value of r, we can calculate:
Time to double = 72 / 0.0241 ≈ 2987.97 years
To complete the table, we can summarize the information as follows:
Initial Investment: $350
Annual % Rate: 2.41% (rounded from 0.0241)
Time to Double: 2987.97 years (rounded)
Therefore, based on continuous compounding, the amount after 10 years is $435.21.