Determine how much time, to the nearest tenth of a year, is required for an investment to triple in value if interest is earned at the rate of 3.5% compounded continuously. (use A=Pe^rt
3=e^3.5t
ln3=3.5t
solve for t.
To determine the time required for an investment to triple in value, we can use the formula for continuous compound interest:
A = P * e^(rt)
Where:
A is the final amount
P is the initial investment
r is the interest rate
t is the time in years
e is Euler's number (approximately 2.71828)
In this case, we want to find the time (t) required for the investment to triple in value. Therefore, the final amount (A) will be three times the initial investment (P).
Let's set up the equation using the given values:
3P = P * e^(0.035t)
Notice that the initial investment (P) will cancel out on both sides of the equation.
3 = e^(0.035t)
To isolate the variable t, we will take the natural logarithm (ln) of both sides of the equation:
ln(3) = ln(e^(0.035t))
Using the logarithmic property, we can bring down the exponent:
ln(3) = 0.035t * ln(e)
Since ln(e) = 1, the equation simplifies to:
ln(3) = 0.035t
Now, we can solve for t by dividing both sides of the equation by 0.035:
t = ln(3) / 0.035
Using a scientific calculator or an online calculator, we can evaluate this expression to find the value of t. The result will be the time required for the investment to triple in years.
Finally, round the answer to the nearest tenth of a year for the final result.