How long does it take for a deposit of $1400 to double at 10% compounded continuously?
To find out how long it takes for a deposit to double at 10% compounded continuously, we can use the continuous compound interest formula:
A = P * e^(rt)
Where:
A = the final amount (in this case, double the deposit, which is $2800)
P = the initial deposit ($1400)
e = the mathematical constant approximately equal to 2.71828
r = the interest rate (10% or 0.10)
t = the time (unknown)
Let's plug in the values and solve for t:
2800 = 1400 * e^(0.10t)
To isolate the exponential term, divide both sides of the equation by 1400:
2 = e^(0.10t)
Next, take the natural logarithm (ln) of both sides to get rid of the exponential:
ln(2) = ln(e^(0.10t))
Using the property of logarithms, we can bring the exponent down to the front:
ln(2) = 0.10t * ln(e)
Since ln(e) is equal to 1, the equation simplifies to:
ln(2) = 0.10t
Now, divide both sides by 0.10 to solve for t:
t = ln(2) / 0.10
Using a calculator, we can find the value of ln(2) to be approximately 0.6931:
t = 0.6931 / 0.10
t ≈ 6.931
Therefore, it takes approximately 6.931 years for a deposit of $1400 to double at a 10% interest rate when compounded continuously.
To find out how long it takes for a deposit to double at a given interest rate compounded continuously, you can use the formula for continuous compound interest:
A = P * e^(rt)
Where:
A = the final amount (in this case, double the initial deposit, $1400 * 2 = $2800)
P = the initial deposit ($1400)
e = the mathematical constant approximately equal to 2.71828
r = the interest rate (in decimal form, 10% = 0.10)
t = the time (in years)
Now, rearrange the formula to solve for t:
t = ln(A/P) / r
ln represents the natural logarithm function.
Let's substitute the values into the formula:
t = ln(2800/1400) / 0.10
Using a calculator or software, we can evaluate this expression:
t ≈ ln(2) / 0.10 ≈ 6.93147 / 0.10 ≈ 69.31 years
Therefore, it would take approximately 69.31 years for a deposit of $1400 to double at a continuous interest rate of 10%.