Suppose you deposit $2000 in an account with an annual interest rate of
7% compounded quarterly. Use the formula A=P(1+r/n)^n*t and round each answer to 2 decimal places, if necessary.
a. Find an equation that gives the amount of money in the account after t
years.
A(t)=
b. Find the amount of money in the account after 8 years.
After 8 years, there will be $ in the account.
c. How many years will it take for the account to contain $4000?
It will take years for there to be
$4000 in the account.
d. If the same account and interest were compounded continuously, how much money would the account contain after 8 years?
With continuous compounding interest, there would be $ in the account after 8 years.
they told you A=P(1+r/n)^n*t
so, for (a) use the numbers they gave you for P,r,n
a. To find the equation that gives the amount of money in the account after t years, we can use the formula A = P(1 + r/n)^(n*t).
In this case, P represents the principal amount (initial deposit), which is $2000. The annual interest rate is 7%, so we convert it to decimal form by dividing it by 100: r = 0.07. The interest is compounded quarterly, so n = 4 (quarterly compounding means 4 times per year).
Now, we can substitute these values into the formula to get the equation:
A(t) = 2000(1 + 0.07/4)^(4*t)
b. To find the amount of money in the account after 8 years, we can substitute t = 8 into the equation A(t):
A(8) = 2000(1 + 0.07/4)^(4*8)
Solving the equation will give us the amount of money in the account after 8 years.
c. To determine how many years it will take for the account to contain $4000, we need to find the value of t when A(t) = 4000. To do this, we set up the equation:
4000 = 2000(1 + 0.07/4)^(4*t)
Then, we solve for t.
d. If the same account and interest were compounded continuously, we would use the formula A = P*e^(r*t), where e is the mathematical constant approximately equal to 2.71828.
Using this formula, we can calculate the amount of money in the account after 8 years:
A(8) = 2000*e^(0.07*8)
Solving this equation will give us the amount of money in the account after 8 years with continuous compounding interest.
a. The equation that gives the amount of money in the account after t years is:
A(t) = P(1 + r/n)^(n*t)
where:
A(t) represents the amount of money in the account after t years
P represents the initial principal (deposit) amount
r represents the annual interest rate (expressed as a decimal)
n represents the number of times interest is compounded per year
t represents the number of years
b. To find the amount of money in the account after 8 years, we can substitute the given values into the formula:
A(8) = 2000(1 + 0.07/4)^(4*8)
A(8) ≈ 2000(1.0175)^(32)
A(8) ≈ 2000(1.640829598) ≈ $3281.66
After 8 years, there will be approximately $3281.66 in the account.
c. To find the number of years it will take for the account to contain $4000, we need to solve the equation:
4000 = 2000(1 + 0.07/4)^(4*t)
Dividing both sides of the equation by 2000 and rearranging, we get:
2 = (1.0175)^(4*t)
Taking the logarithm of both sides, we have:
log(2) = log[(1.0175)^(4*t)]
Using the property log(a^b) = b * log(a), we can simplify further:
log(2) = 4*t * log(1.0175)
Finally, we can solve for t by dividing both sides by 4 * log(1.0175):
t = log(2) / (4 * log(1.0175))
t ≈ 10.24
So, it will take approximately 10.24 years for the account to contain $4000.
d. If the same account and interest were compounded continuously, the formula to calculate the amount of money in the account after t years would be:
A(t) = P * e^(r*t)
where:
e is the mathematical constant approximately equal to 2.71828
Substituting the given values into the formula:
A(8) = 2000 * e^(0.07*8)
A(8) ≈ 2000 * e^(0.56)
A(8) ≈ 2000 * 1.75 ≈ $3500
After 8 years of continuous compounding, the account would contain approximately $3500.