The problem describes a debt to be amortized. (Round your answers to the nearest cent.)

A man buys a house for $\$ 300,000$. He makes a $\$ 150,000$ down payment and amortizes the rest of the purchase price with semiannual payments over the next 8 years. The interest rate on the debt is $13 \%$, compounded semiannually.
(a) Find the size of each payment.
$\$$
(b) Find the total amount paid for the purchase.
$\$$
(c) Find the total interest paid over the life of the loan.
$\$$
Show My Work (Optionai) ?

Step 1 ：\(\text{Loan Amount} = \text{Purchase Price} - \text{Down Payment} = \$300,000 - \$150,000 = \$150,000\)

Step 2 ：\(r = \frac{13\%}{2} = 6.5\% = 0.065\)

Step 3 ：\(n = 8 \times 2 = 16\)

Step 4 ：\(P = \frac{0.065 \times \$150,000}{1 - (1 + 0.065)^{-16}}\)

Step 5 ：\(P = \frac{\$9,750}{1 - (1.065)^{-16}}\)

Step 6 ：\(P = \frac{\$9,750}{0.68433}\)

Step 7 ：\(\boxed{P = \$14,248.06}\)

Step 8 ：\(\text{Total Amount Paid} = \text{Down Payment} + (P \times n) = \$150,000 + (\$14,248.06 \times 16)\)

Step 9 ：\(\boxed{\text{Total Amount Paid} = \$377,968.96}\)

Step 10 ：\(\text{Total Interest Paid} = \text{Total Amount Paid} - \text{Purchase Price} = \$377,968.96 - \$300,000\)

Step 11 ：\(\boxed{\text{Total Interest Paid} = \$77,968.96}\)