If $ 2500 is invested in an account that pays interest compounded continuously, how long will it take to grow to $ 5000 at 3%? Round to the nearest tenth
To find out how long it will take for $2500 to grow to $5000 at 3% interest compounded continuously, we can use the formula:
A = P * e^(rt)
Where:
A = the final amount ($5000)
P = the initial amount ($2500)
e = Euler's number (approximately 2.71828)
r = interest rate per year (3% or 0.03)
t = time in years
Substituting the given values, the formula becomes:
5000 = 2500 * e^(0.03t)
To solve for t, we can divide both sides by 2500:
2 = e^(0.03t)
Next, take the natural logarithm (ln) of both sides of the equation:
ln(2) = ln(e^(0.03t))
Using the property of logarithms, we can simplify as:
ln(2) = 0.03t * ln(e)
Since ln(e) is equal to 1, the equation further simplifies to:
ln(2) = 0.03t
Now, divide both sides by 0.03:
ln(2) / 0.03 = t
Calculating this on a calculator, we find:
t ≈ 23.1
Therefore, it will take approximately 23.1 years for the $2500 to grow to $5000 at 3% interest compounded continuously (rounded to the nearest tenth).
To solve this problem, we can use the formula for compound interest with continuous compounding. The formula is:
A = P * e^(rt)
Where:
A = the final amount
P = the initial principal (or investment)
e = Euler's number (approximately 2.71828)
r = the interest rate
t = the time in years
In this case, we want to find the time it takes for an investment of $2500 to grow to $5000 at an interest rate of 3%.
Plugging in the values into the formula, we have:
5000 = 2500 * e^(0.03t)
Next, we divide both sides of the equation by 2500 to isolate the exponential term:
2 = e^(0.03t)
To solve for t, we can take the natural logarithm of both sides:
ln(2) = ln(e^(0.03t))
Using the property of logarithms ln(a^b) = b * ln(a), the equation simplifies to:
ln(2) = 0.03t * ln(e)
Since ln(e) equals 1, the equation further simplifies to:
ln(2) = 0.03t
To find t, divide both sides of the equation by 0.03:
t = ln(2) / 0.03
Using a calculator, we find:
t ≈ 23.1
Therefore, it will take approximately 23.1 years for the investment of $2500 to grow to $5000 at an interest rate of 3% with continuous compounding.
5000=2500*1.03^t
1.03^t=2
t=ln2/ln1.03=23.4