 Paying Off a Credit Card You owe $8000 on 
a credit card. Because of the high balance, you 
stop charging and begin paying off the card. You 
can afford to make only the minimum monthly 
payment, which is 5% of the balance. The card 
carries an APR of 24%. Under these conditions, 
the remaining balance after t monthly payments is 
given by
B 5 8000 (1.02 3 0.95)t
 dollars.
You feel comfortable resuming charging with 
the card when the balance is between $1000 
and $2000.
a. Plot the graph of B along with the target values 
of 1000 and 2000. Use a horizontal span of 
0 to 100.
b. For what values of t is the function B in the 
desired range?
c. After how many monthly payments can you 
resume charging? (Your answer should be a 
whole number.)  accurate answers and show working 

Here is the graph of the remaining balance $ B(t) $ over time, with the horizontal lines representing the target values of $1000 and $2000. The blue curve shows how the balance decreases over the months.

### Part (b): For what values of $ t $ is $ B(t) $ in the desired range?

We want to find when the balance $ B(t) $ is between $1000 and $2000, i.e., when:

\[
1000 \leq 8000 \cdot (0.969)^t \leq 2000
\]

To solve for $ t $, we’ll work with both bounds.

**Upper bound:**

\[
8000 \cdot (0.969)^t = 2000
\]

Dividing both sides by 8000:

\[
(0.969)^t = \frac{2000}{8000} = 0.25
\]

Taking the natural logarithm of both sides:

\[
t \ln(0.969) = \ln(0.25)
\]

\[
t = \frac{\ln(0.25)}{\ln(0.969)}
\]

**Lower bound:**

\[
8000 \cdot (0.969)^t = 1000
\]

Dividing both sides by 8000:

\[
(0.969)^t = \frac{1000}{8000} = 0.125
\]

Taking the natural logarithm of both sides:

\[
t \ln(0.969) = \ln(0.125)
\]

\[
t = \frac{\ln(0.125)}{\ln(0.969)}
\]

Now, let me calculate these values.The balance $ B(t) $ will be within the desired range of $1000 to $2000 between approximately:

- **44 months** and **66 months**.

### Part (c): After how many monthly payments can you resume charging?

You can resume charging once the balance drops below $2000. From our earlier calculation, this occurs after approximately 44 months.

Thus, you can resume charging after **44 months**.

Would you like any further clarifications or details?

---

Here are 5 related questions for further exploration:
1. How would the results change if the interest rate were different?
2. What happens if you decide to pay more than the minimum payment each month?
3. How is the formula derived from the APR and minimum payment percentage?
4. How much interest would you have paid by the time the balance reaches $2000?
5. What if the initial balance were $10,000 instead of $8,000?

**Tip**: Increasing your monthly payments, even slightly, can significantly reduce the time and interest paid when paying off debt.

Paying Off a Credit Card: You owe $8000 on a credit card. Because of the high balance, you stop charging and begin paying off the card. You can afford to make only the minimum monthly payment, which is 5% of the balance. The card carries an APR of 24%. Under these conditions, the remaining balance after t monthly payments is given by B = 8000 (1.02 * 0.95)^t dollars. You feel comfortable resuming charging with the card when the balance is between $1000 and $2000. a. Plot the graph of B along with the target values of 1000 and 2000. Use a horizontal span of 0 to 100. b. For what values of t is the function B in the desired range? c. After how many monthly payments can you resume charging? (Your answer should be a whole number.)

Explore how long it will take to pay off an $8000 credit card balance with a 24% APR by only making minimum payments. This problem involves exponential decay and finance math. Learn how to calculate when the balance will reach between $1000 and $2000.

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