Find the amount of time required to double a dollar investment at 7.72%, if the interest is compounded continuously.
4.49 years
8.98 years
17.96 years
I'm not understanding this, I got 3.9 years when I tried to solve this.
The basic formula for compounding continuously is:
Lim (1+r/n)^(nt)
as n→∞
where t=number of years, and r is the annual interest rate.
which results in the formula:
A(t)=A(0)e^(rt)
For doubling,
A(t)/A(0)=2
Therefore:
e^(0.0772t) = 2
Take natural log on both sides to get:
0.0772t = ln(2)
Solve for t.
continuous compounding formula is
amount = principal(e^(it)) , so
2 = 1(e^.0772t)
ln2 = ln (e^.0772t)
.0772 lne = ln2 , but ln e = 1
t = ln2/.0772 = 8.98
how did you get 3.9 ?
Well, it seems like you're in quite the pickle. Don't worry, I'm here to help...or at least attempt to help with a little bit of humor!
Let's figure out the time required to double your dollar investment. With continuous compounding and an interest rate of 7.72%, it can get a bit tricky. We use the formula A = P * e^(rt), where A is the final amount, P is the principal (your dollar), e is Euler's number (approximately 2.71828), r is the interest rate, and t is the time in years.
Now, if we set A to 2 (since you want to double your investment) and P to 1 (your initial dollar), we can solve for t. Plugging in the values, we get:
2 = 1 * e^(0.0772t)
To get rid of that pesky exponent, we take the natural logarithm of both sides:
ln(2) = ln(e^(0.0772t))
Using a property of logarithms, we can simplify the equation to:
ln(2) = 0.0772t
Now, divide both sides by 0.0772:
t = ln(2) / 0.0772
Now, let's calculate it:
t ≈ 8.98 years
So, my friend, it seems like you were close! The correct answer is 8.98 years, not 3.9 years. But hey, don't be too hard on yourself. Math likes to sneak in surprises when we least expect them! Keep on smiling and never stop clowning around!
To find the amount of time required to double a dollar investment at 7.72% interest, compounded continuously, you can use the formula for continuous compound interest:
A = P * e^(rt)
Where:
A is the final amount
P is the initial principal amount
r is the annual interest rate (in decimal form)
t is the time in years
e is the mathematical constant approximately equal to 2.71828
In this case, you want to find the time required to double the investment, so A will be 2 times the initial amount (P).
2P = P * e^(0.0772t)
Now, you can cancel out the P on both sides of the equation:
2 = e^(0.0772t)
To isolate t, you need to take the natural logarithm (ln) of both sides of the equation:
ln(2) = ln(e^(0.0772t))
Since ln(e^(0.0772t)) simplifies to 0.0772t, you can rewrite the equation as:
ln(2) = 0.0772t
Now, divide both sides of the equation by 0.0772:
t = ln(2) / 0.0772
Using a calculator, you'll find that ln(2) is approximately 0.6931. Plugging this value into the equation gives us:
t = 0.6931 / 0.0772
t ≈ 8.98 years
So the correct answer is 8.98 years, not 3.9 years.