Suppose that $17,000 is invested in a savings account paying 5.1% 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 3 years if interest is compounded daily. (Round your answer to two decimal places.)
A(3) =
(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.)
yr
A(t) = 17000(1.051)^t
b) i = .051/365
A(t) = 17000(1 + .051/365)^(1095) = $19810.31
continuous ...
20000 = 17000 e^(.051t)
solve for t
(I got 3.1866)
A should be to the ^12t power because it's compounded monthly
(a) The formula for the amount A in the account after t years if interest is compounded monthly can be calculated using the compound interest formula:
A(t) = P(1 + r/n)^(n*t)
where:
A(t) = amount in the account after t years
P = initial investment (principal) = $17,000
r = annual interest rate = 5.1% = 0.051 (in decimal)
n = number of times interest is compounded per year = 12 (for monthly compounding)
t = number of years
So, the formula becomes:
A(t) = 17000(1 + 0.051/12)^(12*t)
(b) To find the amount in the account after 3 years if interest is compounded daily, we need to use the same formula but adjust the compounding frequency. The new formula becomes:
A(3) = 17000(1 + 0.051/365)^(365*3)
Simplifying and calculating this equation will give you the amount in the account after 3 years compounded daily.
(c) To find how long it will take for the amount in the account to grow to $20,000 if interest is compounded continuously, we need to use the continuous compound interest formula:
A(t) = P * e^(r*t)
where:
A(t) = amount in the account after t years
P = initial investment (principal) = $17,000
r = annual interest rate = 5.1% = 0.051 (in decimal)
t = number of years
To find the time it will take for the amount to reach $20,000, we need to solve the equation:
20000 = 17000 * e^(0.051*t)
Solving this equation will give you the amount of time (t) it will take for the amount to grow to $20,000 compounded continuously.
(a) When interest is compounded monthly, we can use the formula for compound interest:
A(t) = P(1 + r/n)^(nt)
Where:
A(t) represents the amount in the account after t years
P represents the principal amount (initial investment)
r represents the annual interest rate (as a decimal)
n represents the number of times interest is compounded per year
t represents the number of years
In this case, the initial investment (principal) is $17,000, the annual interest rate (r) is 5.1% (or 0.051 as a decimal), and interest is compounded monthly (so n = 12 since there are 12 months in a year).
Therefore, the formula for the amount A in the account after t years if interest is compounded monthly is:
A(t) = 17000(1 + 0.051/12)^(12t)
(b) To find the amount in the account after 3 years if interest is compounded daily, we can use the same formula, but with a different value for n. Since interest is compounded daily, we have n = 365 (since there are 365 days in a year).
Plugging in the values into the formula:
A(3) = 17000(1 + 0.051/365)^(365*3)
Using a calculator or a computer program, calculate the value of A(3) to two decimal places.
(c) To determine how long it will take for the amount in the account to grow to $20,000 if interest is compounded continuously, we need to use the formula for continuous compound interest:
A(t) = P * e^(rt)
Where:
A(t) represents the amount in the account after t years
P represents the principal amount (initial investment)
r represents the annual interest rate (as a decimal)
t represents the number of years
e is Euler's number, a mathematical constant approximately equal to 2.71828
In this case, the initial investment (principal) is $17,000, the target amount is $20,000, and the annual interest rate (r) is 5.1% (or 0.051 as a decimal).
So, the formula becomes:
20000 = 17000 * e^(0.051t)
We need to solve for t, so divide both sides of the equation by 17,000 and take the natural logarithm (ln) of both sides:
ln(20000/17000) = ln(e^(0.051t))
ln(20000/17000) = 0.051t
Finally, divide both sides by 0.051 to solve for t:
t = ln(20000/17000) / 0.051
Calculate the value of t using a calculator or a computer program, and round it to two decimal places.