You have a credit card with a balance of $3000. Each month the minimum payment is 2.5% of the balance. The monthly interest rate is 1.5% of the balance. You decide you would like to pay off this card by making the minimum payment each month. You cut up the card so that you cannot change your mind and use it again.
a) Find a decreasing periodic exponential function that models the balance at the beginning of each month.
b) How many months will it take to get the balance down to $5.00?
To find the decreasing periodic exponential function that models the balance at the beginning of each month, we need to consider the factors that affect the balance: the minimum payment and the monthly interest rate.
a) Let's break down the information given:
- Initial balance: $3000
- Minimum payment: 2.5% of the balance
- Monthly interest rate: 1.5% of the balance
Now, let's use this information to calculate the balance at the beginning of each month:
1) Calculate the minimum payment:
Minimum payment = 2.5% of $3000 = $75
2) Calculate the interest for the month:
Interest for the month = 1.5% of $3000 = $45
3) Subtract the minimum payment and the interest from the balance:
Balance at the beginning of the next month = Current balance - Minimum payment - Interest
Balance at the beginning of the next month = $3000 - $75 - $45 = $2880
So, the balance at the beginning of the second month is $2880.
To find the balance at the beginning of each subsequent month, follow the same steps:
1) Calculate the minimum payment:
Minimum payment = 2.5% of previous balance
2) Calculate the interest for the month:
Interest for the month = 1.5% of previous balance
3) Subtract the minimum payment and the interest from the balance:
Balance at the beginning of the next month = Previous balance - Minimum payment - Interest
Repeat these steps for each month to find the balance at the beginning of each month.
b) To determine the number of months it will take to get the balance down to $5.00, we need to continue the process until the balance becomes less than or equal to $5.00.
Using the same steps as before, calculate the balance at the beginning of each month until it reaches $5.00 or below. Count the number of months it takes to reach that point.
a) To find a decreasing periodic exponential function that models the balance at the beginning of each month, we need to consider the minimum payment and the monthly interest rate.
Let's define the balance at the beginning of each month as B(n), where n represents the month. We know that the minimum payment is 2.5% of the current balance, and the monthly interest rate is 1.5% of the current balance.
Since the minimum payment is 2.5% of the balance, we have: Minimum payment = 0.025 * B(n)
And since the monthly interest rate is 1.5% of the balance, we have: Monthly interest = 0.015 * B(n)
So, the balance at the beginning of each month can be represented by the following equation:
B(n + 1) = B(n) - (Minimum payment) + (Monthly interest)
Substituting the expressions for the minimum payment and monthly interest, we have:
B(n + 1) = B(n) - 0.025 * B(n) + 0.015 * B(n)
Simplifying the equation, we get:
B(n + 1) = B(n) * (1 - 0.025 + 0.015)
Or,
B(n + 1) = B(n) * 0.99
This is the decreasing periodic exponential function that models the balance at the beginning of each month.
b) To find out how many months it will take to get the balance down to $5.00, we can set up the equation:
B(n) = $5.00
Substituting the equation B(n + 1) = B(n) * 0.99, we get:
B(n) * 0.99 = $5.00
Dividing both sides by 0.99, we have:
B(n) = $5.00 / 0.99
Calculating the value on the right-hand side, we find:
B(n) ≈ $5.05
Therefore, it will take approximately n months to get the balance down to $5.00.