a)Find the time required to double the amount of an investment at an interest rate k, compounded continuously.
b) Find the time required to double an investment at continuous rate of 6%.
e^kt = 2
t = ln2/k
now plug in 6%
a) To find the time required to double an investment at an interest rate, you can use the formula for continuous compounding:
A = P*e^(kt)
Where:
A = Amount you want to obtain (in this case, double the initial investment)
P = Initial investment
k = Interest rate
t = Time (in years)
Since you want to double the initial investment, the final amount (A) will be 2 times the initial investment (P). So we can rewrite the formula as:
2P = P*e^(kt)
To solve for t, divide both sides of the equation by P:
2 = e^(kt)
Next, take the natural logarithm (ln) of both sides of the equation:
ln(2) = ln(e^(kt))
Using the property of logarithms (ln(e^x) = x), simplify the equation to:
ln(2) = kt
Finally, solve for t by dividing both sides of the equation by k:
t = ln(2) / k
This formula will give you the time required to double the investment at an interest rate k, compounded continuously.
b) To find the time required to double an investment at a continuous rate of 6%, substitute the value of k into the formula we derived earlier.
k = 6% = 0.06 (since 6% is equivalent to 0.06 as a decimal)
t = ln(2) / 0.06
Using a calculator, find the natural logarithm of 2 (ln(2)), then divide the result by 0.06 to get the time required to double the investment at a continuous rate of 6%.