fiona invested $1000 at 6% compounded continuously. at the same time, maria invested $1100 at 6% compounded daily. how long will it take for their investments to be equal in value?
step by step please!
we want
1000e^.06x = 1100(1+.06/365)^(365x)
x = 202,800 years
as for solving the equation, there's no useful symbolic method. Some kind of numeric approximation will probably be needed.
The time seems rather large. Probably because daily compounding is very nearly continuous, and 1100 gets a pretty big head start over 1000.
You sure the interest rates are the same?
To find out how long it will take for Fiona and Maria's investments to be equal in value, we can set up an equation based on the compound interest formula.
The compound interest formula is given by:
A = P * e^(rt)
Where:
A = the future value of the investment
P = the principal amount (initial investment)
e = Euler's number (approximately 2.71828)
r = the interest rate per period
t = the time in years
We want to find the time (t) it takes for their investments to be equal, so we'll set up two equations and solve for t.
For Fiona's investment:
A1 = 1000 * e^(0.06t)
For Maria's investment:
A2 = 1100 * (1 + (0.06/365))^(365t)
Since we want to find when both investments are equal, we'll set A1 equal to A2:
1000 * e^(0.06t) = 1100 * (1 + (0.06/365))^(365t)
Now we can solve for t. However, solving this equation analytically may not be straightforward. We can use numerical methods like trial and error or a graphing calculator to find an approximate solution.
For trial and error, we can start by guessing a value for t and checking if the equation is satisfied. If not, we can adjust our guess accordingly until we find a value that satisfies the equation.
Alternatively, we can use technology to graph the functions A1 and A2 on a graphing calculator or online graphing tool, and find the point of intersection. This point will indicate when their investments are equal.
Either method will provide an approximate solution for the time it takes for their investments to be equal.