How many years and days will it take for a deposit of $700 to double at 4% interest compounded continuously?
1400 = 700 e^(.04t)
2 = e^(.04t)
ln 2 = ln (e^(.04t))
ln 2 = .04t
t = ln /.04 = 17.33 years
= 17 years and appr. 120 days
To determine how many years and days it will take for a deposit to double at 4% interest compounded continuously, you can use the continuous compound interest formula:
A = P * e^(rt)
Where:
A is the future amount or balance
P is the principal amount (initial deposit)
e is the base of the natural logarithm (approximately 2.71828)
r is the interest rate per period (in decimal form)
t is the time in years
In this case, we want to find the time (t), so we'll rearrange the formula:
t = ln(A/P) / r
Let's plug in the values:
P = $700 (initial deposit)
A = 2 * P = $1,400 (double the initial deposit)
r = 4% = 0.04 (interest rate in decimal form)
Now, substitute these values into the formula and calculate:
t = ln(1400/700) / 0.04
t = ln(2) / 0.04
Using a calculator, we can find the natural logarithm of 2 is approximately 0.6931:
t ≈ 0.6931 / 0.04
t ≈ 17.3277
Since we're looking for the time in years and days, we'll round down the number of years to 17. The remaining decimal represents the fraction of the year, which we can convert to days by multiplying it by 365:
0.3277 * 365 ≈ 119.4745
Rounding down, we find that it will take approximately 17 years and 119 days for a deposit of $700 to double at a 4% interest rate compounded continuously.
To calculate the number of years and days it will take for a deposit to double at a given interest rate compounded continuously, we can use the formula:
t = ln(2) / (r * ln(e))
Where:
t = time in years
ln = natural logarithm
r = interest rate
e = Euler's number (approximately 2.71828)
In this case, the deposit is $700 and the interest rate is 4%, so we can substitute the values into the formula:
t = ln(2) / (0.04 * ln(2.71828))
Now, let's calculate the value:
t ≈ ln(2) / (0.04 * 0.999628624)
Using a scientific calculator or an online calculator, we can find that ln(2) ≈ 0.693147181, and perform the calculation:
t ≈ 0.693147181 / 0.039985145
t ≈ 17.33123116
So, it would take approximately 17.33 years for a deposit of $700 to double at a 4% interest rate compounded continuously.
To calculate the number of days, we can multiply the decimal part of the years by the number of days in a year (365):
days ≈ 0.33 * 365
days ≈ 120.45
Therefore, it would take approximately 17 years and 120 days for the deposit to double at a 4% interest rate compounded continuously.