Answer:
Equation is A = 5000(1.015)^(4t)
Her investment will be worth $10,000 in about <u>11.63888 years</u>
Rounding up to the nearest whole number gets to <u>12 years</u>
==========================================================
Explanation:
Part 1) Finding the equation
The compound interest formula is
A = P(1+r/n)^(n*t)
Here are the variables
- A = final amount
- P = starting amount, or deposit, or principal
- r = interest rate in decimal form
- n = number of times money is compounded per year
- t = number of years
In this case,
- P = 5000
- r = 0.06 from the 6% annual interest
- n = 4 times a year is the compounding frequency
- t = unknown amount of time
Therefore, the equation is
A = P(1+r/n)^(n*t)
A = 5000(1+0.06/4)^(4t)
A = 5000(1.015)^(4t)
The decimal value is exact.
--------------------------
Part 2) Let's plug in A = 10,000 and solve for t.
You'll need to use logarithms to isolate the exponent.
A = 5000(1.015)^(4t)
10,000 = 5000(1.015)^(4t)
10,000/5000 = (1.015)^(4t)
2 = (1.015)^(4t)
Log[ 2 ] = Log[ (1.015)^(4t) ]
Log(2) = 4t*Log( 1.015 )
4t = Log(2)/Log(1.015)
4t = 46.5555256308062
t = 46.5555256308062/4
t = 11.6388814077015
t = 11.63888
It takes about 11.63888 years for the investment to reach $10,000.
Therefore, at the 12 year mark is when the investment is more than $10,000.
