You deposit $9000 in a savings account that earns 3.6% annual interest compounded monthly. You also save $40 per month in a safe at home. Write a function C(t) = b(t) + h(t) where b(t) represents the balance of your savings account and h(t) represents the amount in your safe after t years. What does c(t) represent? What is the function?
9000(1.003)^12t+40*12t
3.6 /12 = monthly interest rate = r
b(t) = 9000 (1 + r)^12t
c(t) = 40*12 t
C(t) = 9000 (1 + r)^12t + 40*12 t
C(t) represents the total balance of your savings, including both the amount in your savings account (b(t)) and the amount in your safe at home (h(t)). The function C(t) can be written as:
C(t) = b(t) + h(t)
Now let's break down the individual components of the function:
1. b(t) represents the balance of your savings account after t years. Since it's compounded monthly at an annual interest rate of 3.6%, we can use the formula for compound interest:
b(t) = P(1 + r/n)^(nt)
Where:
P = principal amount (initial deposit) = $9000
r = annual interest rate = 3.6% = 0.036
n = number of times interest is compounded per year = monthly = 12
t = number of years
Plugging in the values, we get:
b(t) = 9000(1 + 0.036/12)^(12t)
2. h(t) represents the amount in your safe after t years, considering a monthly contribution of $40. Since the amount in your safe is increasing linearly over time, we can use the formula for compound interest:
h(t) = P + (n * c * t)
Where:
P = principal amount (initial amount in the safe) = 0 (since there was no initial amount)
n = number of monthly contributions = t * 12
c = monthly contribution amount = $40
t = number of years
Plugging in the values, we get:
h(t) = 0 + (t * 12 * 40)
Therefore, the final function C(t) is:
C(t) = 9000(1 + 0.036/12)^(12t) + (t * 12 * 40)
The function C(t) represents the total balance of your savings after t years. It is given by the sum of two functions: b(t) represents the balance in your savings account, and h(t) represents the amount in your safe.
To find the function for b(t), we need to consider the compound interest on your savings account. The formula for compound interest is:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment/loan, including interest
P = the principal investment amount (initial deposit)
r = annual interest rate (as a decimal)
n = number of times that interest is compounded per year
t = number of years the money is invested/borrowed for
In this case, P = $9000, r = 3.6% (or 0.036 as a decimal), n = 12 (compounded monthly), and t is the number of years.
So, the function for b(t) is:
b(t) = 9000(1 + 0.036/12)^(12t)
To find the function for h(t), we need to consider the amount saved in your safe each month. Since you save $40 per month, after t years, you will have saved:
h(t) = 40 * 12t
Now, to find the function C(t), we simply sum the two functions:
C(t) = b(t) + h(t) = 9000(1 + 0.036/12)^(12t) + 40 * 12t
So, the function C(t) represents the total balance of your savings (including both the savings account and the safe) after t years.