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 note that the annual percentage rate is
$$
\text{APR} = \text{prime rate} + 4\% = 6\% + 4\% = 10\%.
$$
Interest is charged daily so the daily rate is
$$
r = \frac{10\%}{365} = \frac{0.10}{365}.
$$

The account history is as follows:

1. **October 1–3:**  
   The starting balance on October 1 is
   $$
B_1 = \$9{,}300.
   $$
   No transactions occur until October 4, so interest accrues for 3 days (October 1, 2, and 3). The interest for this period is:
   $$
I_1 = B_1 \times r \times 3 = 9300 \times \frac{0.10}{365} \times 3.
   $$

2. **October 4–12:**  
   On October 4 a payment of
   $$
P = \$1{,}800
   $$
   is made. The new balance immediately becomes:
   $$
B_2 = 9300 - 1800 = \$7{,}500.
   $$
   This balance accrues interest for 9 days (October 4 through October 12). The interest for this period is:
   $$
I_2 = B_2 \times r \times 9 = 7500 \times \frac{0.10}{365} \times 9.
   $$

3. **October 13–18:**  
   On October 13 the business borrows 
   $$
L_1 = \$2{,}900.
   $$
   The balance then becomes:
   $$
B_3 = 7500 + 2900 = \$10{,}400.
   $$
   This balance accrues interest for 6 days (October 13 through October 18). The interest for this period is:
   $$
I_3 = B_3 \times r \times 6 = 10400 \times \frac{0.10}{365} \times 6.
   $$

4. **October 19–31:**  
   On October 19 the business borrows 
   $$
L_2 = \$4{,}400.
   $$
   The new balance becomes:
   $$
B_4 = 10400 + 4400 = \$14{,}800.
   $$
   This balance accrues interest for 13 days (October 19 through October 31). The interest for this period is:
   $$
I_4 = B_4 \times r \times 13 = 14800 \times \frac{0.10}{365} \times 13.
   $$

Let us compute each interest component:

1. For the first period:
   $$
I_1 = 9300 \times \frac{0.10}{365} \times 3 = 9300 \times \frac{0.30}{365} \approx 9300 \times 0.00082192 \approx \$7.64.
   $$

2. For the second period:
   $$
I_2 = 7500 \times \frac{0.10}{365} \times 9 = 7500 \times \frac{0.90}{365} \approx 7500 \times 0.00246575 \approx \$18.49.
   $$

3. For the third period:
   $$
I_3 = 10400 \times \frac{0.10}{365} \times 6 = 10400 \times \frac{0.60}{365} \approx 10400 \times 0.00164384 \approx \$17.11.
   $$

4. For the fourth period:
   $$
I_4 = 14800 \times \frac{0.10}{365} \times 13 = 14800 \times \frac{1.30}{365} \approx 14800 \times 0.00356164 \approx \$52.69.
   $$

Now, add all the interest amounts:
$$
I_{\text{total}} = I_1 + I_2 + I_3 + I_4 \approx 7.64 + 18.49 + 17.11 + 52.69 \approx \$95.93.
$$

The ending principal balance after all transactions (before interest) is:
$$
\text{Principal} = 9300 - 1800 + 2900 + 4400 = 9300 - 1800 + (2900 + 4400) = 7500 + 7300 = \$14{,}800.
$$

Finally, adding the interest:
$$
\text{New Balance} = 14800 + 95.93 = \$14{,}895.93.
$$

Thus, the new balance is 
$$
\boxed{\$14{,}895.93}.
$$
The new balance is \$14,895.93.

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