To determine how long it will take for Fiona and Maria's investments to be equal in value, we need to set up an equation using the formula for continuous compound interest and the formula for daily compound interest.
The formula for continuous compound interest is:
A = P * e^(rt)
where:
A = final amount
P = principal (initial investment)
e = base of natural logarithms (approximately 2.71828)
r = interest rate (as a decimal)
t = time (in years)
The formula for daily compound interest is:
A = P * (1 + r/n)^(n*t)
where:
A = final amount
P = principal (initial investment)
r = interest rate (as a decimal)
n = number of times interest is compounded per year
t = time (in years)
Let's solve the problem step by step:
Step 1: Set up the equation for Fiona's investment.
Fiona's investment is $1000 at 8% compounded continuously.
So, A_f = 1000 * e^(0.08t)
Step 2: Set up the equation for Maria's investment.
Maria's investment is $1100 at 8% compounded daily.
So, A_m = 1100 * (1 + 0.08/365)^(365*t)
Step 3: Set the two equations equal to each other.
1000 * e^(0.08t) = 1100 * (1 + 0.08/365)^(365*t)
Step 4: Simplify the equation.
Divide both sides of the equation by 1000 and 1100 to simplify:
e^(0.08t) = (1 + 0.08/365)^(365*t) * 11/10
Step 5: Solve for t numerically.
To solve this equation numerically, you can use a graphing calculator or a computational software like Python or Wolfram Alpha. The goal is to find the value of t when both sides of the equation are equal.
Using a calculator or software, you can plot the graphs of y = e^(0.08x) and y = (1 + 0.08/365)^(365*x) * 11/10. The point where the two graphs intersect is the solution for t.
Alternatively, if you are using Python, you can use the `fsolve` function from the `scipy.optimize` library to find the solution numerically.
Step 6: Calculate the approximate time it takes for their investments to be equal.
Once you have the numerical solution for t, you can calculate the time it takes for Fiona and Maria's investments to be equal by substituting the value of t from step 5 into either of the original equations (A_f or A_m).
Note: The precise numerical value for t will depend on your chosen method for solving the equation.