Suppose that $11,000 is invested in a savings account paying 5.5% interest per year:

Question

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.

Answer 1

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*e^rt

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