Fiona invested
$
3000
at
6
%
compounded continuously. At the same time, Maria invested
$
3200
at
6
%
compounded daily. How long will it take (to the nearest day) for their investments to be equal in value?
why all those newlines? Very annoying.
3000e^(.06x) = 3200(1 + .06/365)^(365x)
x = 13,088.4 years
daily and continuous compounding are very close to equal, so that $200 head start takes a long time to make up.
To find out how long it will take for Fiona's and Maria's investments to be equal in value, we can use the formula for continuous compound interest:
A = P * e^(rt)
where:
A = the future value of the investment
P = the principal amount (initial investment)
e = the mathematical constant approximately equal to 2.71828
r = the annual interest rate (in decimal form)
t = the time period in years
For Fiona's investment:
P = $3000
r = 6% = 0.06 (converted to decimal)
A_fiona = $3000 * e^(0.06t)
For Maria's investment:
P = $3200
r = 6% = 0.06 (converted to decimal)
A_maria = $3200 * (1 + (0.06/365))^(365t)
We want to find the time (t) when their investments are equal, so:
3000 * e^(0.06t) = 3200 * (1 + (0.06/365))^(365t)
To solve this equation, we can use trial and error or use a graphing calculator. Since you wanted the answer to the nearest day, let's use a graphing calculator.
Step1: Graph the two functions on the calculator:
y1 = 3000 * e^(0.06x)
y2 = 3200 * (1 + (0.06/365))^(365x)
Step 2: Find the point of intersection of the two graphs.
Step 3: Read the x-coordinate (t-value) of the intersection point. This will give us the time it takes for their investments to be equal.
Please note that since the values are rounded, the final answer may vary slightly.