Cook Security Systems has a $$ \$ 37,500 $$ line of credit, which charges an annual percentage rate of prime rate plus $$ 4 \% $$. The starting balance on October 1 was $$ \$ 9,300 $$. On October 4 they made a payment of $$ \$ 1,800 $$. On October 13 the business borrowed $$ \$ 2,900 $$, and on October 19 they borrowed $$ \$ 4,400 $$. If the current prime rate is $$ 6 \% $$, what is the new balance (in $$ \$ $$ )? (Round your answer to the nearest cent.)

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We first determine the interest‐bearing periods and the balance during each period. The annual interest rate is

$$
\text{APR} = \text{prime} + 4\% = 6\% + 4\% = 10\%.
$$

Since interest accrues daily, the daily interest rate is

$$
r_{\text{daily}} = \frac{0.10}{365}.
$$

The transactions during October are as follows:

1. **From October 1 to October 3 (3 days):**  
   The starting balance is $$\$9,\!300$$.

2. **On October 4:**  
   A payment of $$\$1,\!800$$ is made.  
   New balance becomes
   $$
   \$9,\!300 - \$1,\!800 = \$7,\!500.
   $$
   This balance remains from October 4 through October 12 (9 days).

3. **On October 13:**  
   The business borrows $$\$2,\!900$$.  
   New balance becomes
   $$
   \$7,\!500 + \$2,\!900 = \$10,\!400.
   $$
   This balance is in effect from October 13 through October 18 (6 days).

4. **On October 19:**  
   They borrow an additional $$\$4,\!400$$.  
   New balance becomes
   $$
   \$10,\!400 + \$4,\!400 = \$14,\!800.
   $$
   This balance remains from October 19 through October 31 (13 days).

Next, we compute the total “balance–days” by multiplying each balance by the number of days it was in effect:

- Period 1 (3 days):  
  $$
  9,\!300 \times 3 = 27,\!900.
  $$
- Period 2 (9 days):  
  $$
  7,\!500 \times 9 = 67,\!500.
  $$
- Period 3 (6 days):  
  $$
  10,\!400 \times 6 = 62,\!400.
  $$
- Period 4 (13 days):  
  $$
  14,\!800 \times 13 = 192,\!400.
  $$

The sum of these balance–days is

$$
27,\!900 + 67,\!500 + 62,\!400 + 192,\!400 = 350,\!200.
$$

Since October has 31 days, the interest accrued over October is calculated by multiplying the total balance–days by the daily interest rate:

$$
\text{Interest} = 350,\!200 \times \frac{0.10}{365}.
$$

Evaluating the daily rate:

$$
\frac{0.10}{365} \approx 0.00027397,
$$

so

$$
\text{Interest} \approx 350,\!200 \times 0.00027397 \approx 95.80.
$$

Thus, about $$\$95.80$$ in interest accrues during October.

Finally, the new balance at the end of October is the balance from the last transaction plus the accrued interest:

$$
\text{New Balance} = \$14,\!800 + \$95.80 \approx \$14,\!895.80.
$$

So, the new balance is approximately $$\$14,\!895.80$$.
The new balance is approximately \$14,895.80.

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