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Suppose that $11,000 is invested in a savings account paying 5.5% interest per year:
(a). Write the formula for the amount A in the account after t years if interest is compounded monthly.
A(t) = __________________
(b). Find the amount in the account after 2 years if interest is compounded daily. (Round answer two decimal places).
(c) How long will it take for the amount in the account to grow to $20,000 if interest is compounded continuously? (Round your answer to two decimal places).
Thank you for your help.
Solution
To begin we will use the compound interest formula: A= P (1 + r/n)(nt),
where P= initial amount, A= final amount, r= rate n= number of payments per time period t= number of time periods elapsed
we are given an intial amount of $11,000 so our P= $11,000
we are given a rate of 5.5% interest, so our r (which is always in decimal form) = 0.055
(a) We are asked to write the equation if the interest was compounded monthly over t years
So, our t= number of years passed
and n= number of payments per year, which would be 12 in this case, because there is a payment once every month.
A= P (1 + r/n)(nt)
A= $11,000 (1 + 0.055/12) (12 * t)
(b) Now we are actually given the amount of time in years, t, but the interest is now compounded DAILY, so there is a payment every day during one given year. So;
t= 2 years
n= every day ; 365 days per year, 365 payments per year
A= $11,000 (1 + 0.055/365) (365 * 2)
now we just need to evaluate the expression
= 11,000 (1.000150685)730
= 11,000 * 1.1162688 = $12,278.96
(c) how long will it take = we are solving for t this time
This interest, however, is compounded continuously so we will use the formula
A = P*ert
We are asked how long it will take for the account to reach $20,000, so $20,000 is our final amount or A.
20,000 = 11,000*e(0.055t)
divide equation by 11,000
take ln of both sides to bring down exponent
ln(20,000 / 11,000 )= 0.055t
divide equation by 0.055
ln(20,000 / 11,000 ) / (0.055) = t
t = 10.86976 , 10.87 years