Question
zeenthezine
If P dollars is invested in a savings account that pays 5 percent annual interest, compounded continuously, in how many years will the account value be equal to 2P dollars?
If P dollars is invested in a savings account that pays 5 percent annual interest, compounded continuously, in how many years will the account value be equal to 2P dollars?
Answers
GPT 3.5
To solve this problem, we can use the formula for compound interest:
A = P * e^(rt)
Where:
A is the final account value
P is the initial principal (dollars)
e is the mathematical constant approximately equal to 2.71828
r is the annual interest rate (decimal form)
t is the time in years
In this case, we want to find the time it takes for the account value to be equal to 2P dollars. So we have:
2P = P * e^(0.05t)
Divide both sides of the equation by P:
2 = e^(0.05t)
Take the natural logarithm (ln) of both sides:
ln(2) = 0.05t * ln(e)
Since ln(e) equals 1, the equation simplifies to:
ln(2) = 0.05t
Now we can divide both sides of the equation by 0.05:
ln(2) / 0.05 = t
Using a calculator, we find:
t ≈ 13.86
Therefore, it will take approximately 13.86 years for the account value to be equal to 2P dollars.
A = P * e^(rt)
Where:
A is the final account value
P is the initial principal (dollars)
e is the mathematical constant approximately equal to 2.71828
r is the annual interest rate (decimal form)
t is the time in years
In this case, we want to find the time it takes for the account value to be equal to 2P dollars. So we have:
2P = P * e^(0.05t)
Divide both sides of the equation by P:
2 = e^(0.05t)
Take the natural logarithm (ln) of both sides:
ln(2) = 0.05t * ln(e)
Since ln(e) equals 1, the equation simplifies to:
ln(2) = 0.05t
Now we can divide both sides of the equation by 0.05:
ln(2) / 0.05 = t
Using a calculator, we find:
t ≈ 13.86
Therefore, it will take approximately 13.86 years for the account value to be equal to 2P dollars.
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