Approximately how long will it take an investment of $300 to triple, if the interest rate is 3.5% per year compounded continuously?
Formula: A(t)=Pe^rt
A=3*300=900
r=3.5%=0.035
P=300
=> 900=300e^0.035t
=>3=e^0.035t
=>ln3=lne^0.035t
=>1.09=0.035t
=>t=1.09/0.035=31 yeears
correct
To find out how long it will take for an investment of $300 to triple at an interest rate of 3.5% per year compounded continuously, we can use the formula for continuous compound interest:
A = P * e^(rt)
Where:
A = Future value (triple the initial investment)
P = Principal amount (initial investment)
e = Euler's number (approximately 2.71828)
r = Annual interest rate (as a decimal)
t = Time (unknown)
Since we want the investment to triple, the future value (A) will be 3 times the initial investment (P), i.e., A = 3P.
Substituting the known values into the formula, we get:
3P = P * e^(0.035t)
Dividing both sides by P, we have:
3 = e^(0.035t)
Next, take the natural logarithm (ln) of both sides to isolate the exponent:
ln(3) = 0.035t * ln(e)
Since ln(e) is equal to 1, this simplifies to:
ln(3) = 0.035t
Now, divide both sides by 0.035 to solve for t:
t = ln(3)/0.035
Using a calculator, we find:
t ≈ 19.8439
Therefore, it will take approximately 19.8439 years for the investment of $300 to triple at an interest rate of 3.5% per year compounded continuously.
To calculate the approximate time it will take for an investment to triple with a continuous compounding interest rate, we can use the formula for continuous compound interest:
A = P * e^(rt)
Where:
A = ending amount
P = principal amount (initial investment)
e = Euler's number (approximately 2.71828)
r = interest rate per period (in this case, 3.5% per year, or 0.035)
t = time in years
In this case, we want to find the time (t) needed for the investment to triple, so the ending amount (A) will be 3 times the principal amount (P).
A = 3P
Substituting these values into our formula:
3P = P * e^(0.035t)
Dividing both sides by P:
3 = e^(0.035t)
To solve for t, we need to take the natural logarithm of both sides:
ln(3) = 0.035t
Now, we can solve for t by dividing both sides by 0.035:
t = ln(3) / 0.035
Using a scientific calculator or math software, we can calculate the approximate value of t:
t ≈ 19.8902 years
Therefore, it will take approximately 19.8902 years for the investment of $300 to triple at an interest rate of 3.5% per year compounded continuously.