Write a function that represents the situation. Find the balance A in the account after the given time period t.
$2000 deposit that earns 5% annual interest compounded quarterly; 5 years
A function is A=
The balance of the account after 5 years is $
A = 2000(1 + 0.05/4)^(4*5) = _____
40500
To find the balance A in the account after the given time period t, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the balance in the account after time t
P = the amount of the deposit
r = the annual interest rate (as a decimal)
n = the number of times the interest is compounded per year
t = the time period in years
In this situation, the deposit is $2000, the annual interest rate is 5%, and the interest is compounded quarterly (n = 4). The time period is 5 years (t = 5).
Plugging in the values into the formula:
A = 2000(1 + 0.05/4)^(4*5)
Simplifying the calculation:
A = 2000(1.0125)^(20)
Calculating the power:
A = 2000 * 1.2762815625
Calculating the final balance:
A ≈ 2552.56
Therefore, the balance of the account after 5 years is $2552.56.
The function representing this situation is A = 2000(1 + 0.05/4)^(4*t).
To find the balance A in the account after the given time period t, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
where:
A - final balance after time period t
P - principal amount (initial deposit)
r - annual interest rate (in decimal form)
n - number of times interest is compounded per year
t - time period in years
In this case, we have a $2000 deposit that earns 5% annual interest compounded quarterly over a period of 5 years. Hence, we need to calculate A using the above formula.
Substituting the given values into the formula, we have:
P = $2000
r = 5% = 0.05 (since the rate is given as a percentage, we need to convert it to decimal form)
n = 4 (compounded quarterly)
t = 5 years
Plugging in the values:
A = 2000 * (1 + 0.05/4)^(4 * 5)
Now we can simplify the expression inside the parentheses first:
A = 2000 * (1 + 0.0125)^(4 * 5)
Next, we calculate the value inside the parentheses:
A = 2000 * (1.0125)^(20)
Finally, we calculate the exponential expression:
A ≈ 2000 * 1.283676
Therefore, the balance A in the account after 5 years would be approximately $2,567.35.