You want to make an investment in a continuously compounding account over a period of 20 years. What interest rate is required for your investment to double in that time period? Round the logarithm value to the nearest hundredth and the answer to the nearest tenth.You want to make an investment in a continuously compounding account over a period of 20 years. What interest rate is required for your investment to double in that time period? Round the logarithm value to the nearest hundredth and the answer to the nearest tenth
2 = e^(r * 20)
ln(2) = 20 r
P = Po*e^rt = 2Po.
Po*e^rt = 2Po.
e^rt = 2,
rt*Lne = Ln2,
r*t = Ln2/Lne = 0.693,
r = 0.693/t = 0.693/20 = 0.035 = 3.5 %.
To find the interest rate required for an investment to double in a continuously compounding account over a period of 20 years, we can use the formula for compound interest:
A = P * e^(rt)
Where:
A = Final amount (double the initial investment)
P = Principal amount (initial investment)
e = Euler's number (approximately 2.71828)
r = Interest rate (unknown)
t = Time period (20 years)
Since we want the investment to double, the final amount will be 2 times the initial investment amount:
2P = P * e^(rt)
We can cancel out the common factor of P from both sides:
2 = e^(rt)
Next, let's take the natural logarithm (ln) of both sides of the equation to solve for r:
ln(2) = ln(e^(rt))
Using the logarithmic property ln(e^x) = x:
ln(2) = rt * ln(e)
Since ln(e) is the natural logarithm of Euler's number (e), it is equal to 1:
ln(2) = rt
Now, we can solve for r by isolating it:
r = ln(2) / t
Substituting the values, we have:
r = ln(2) / 20
Calculating the value, we get:
r ≈ 0.03466
Rounding the logarithm value to the nearest hundredth and the answer to the nearest tenth, the interest rate required for the investment to double in 20 years is approximately 3.5%.