Question: The owner of a construction company purchased a cement mixer truck for $160,000 and made a down payment of $45,000. The remaining balance was financed for 10 years at an annual interest rate of 7.75% compounded monthly. Find the monthly payment (in dollars). (Round your answer to the nearest cent.)
1. **State the problem:**
We need to find the monthly payment for a loan where the principal is the remaining balance after a down payment, the interest rate is compounded monthly, and the loan term is 10 years.
2. **Identify the known values:**
- Purchase price: $160,000$
- Down payment: $45,000$
- Loan amount (principal) $P = 160,000 - 45,000 = 115,000$
- Annual interest rate $r = 7.75\% = 0.0775$
- Monthly interest rate $i = \frac{0.0775}{12}$
- Number of payments $n = 10 \times 12 = 120$
3. **Formula for monthly payment on an amortized loan:**
$$
M = P \times \frac{i(1+i)^n}{(1+i)^n - 1}
$$
where $M$ is the monthly payment.
4. **Calculate monthly interest rate:**
$$
i = \frac{0.0775}{12} = 0.0064583333
$$
5. **Calculate $(1+i)^n$:**
$$
(1 + 0.0064583333)^{120} \approx 2.12749
$$
6. **Substitute values into the formula:**
$$
M = 115,000 \times \frac{0.0064583333 \times 2.12749}{2.12749 - 1}
$$
7. **Calculate numerator:**
$$
0.0064583333 \times 2.12749 = 0.01374
$$
8. **Calculate denominator:**
$$
2.12749 - 1 = 1.12749
$$
9. **Calculate fraction:**
$$
\frac{0.01374}{1.12749} \approx 0.01218
$$
10. **Calculate monthly payment:**
$$
M = 115,000 \times 0.01218 = 1,400.70
$$
**Final answer:** The monthly payment is approximately $\boxed{1400.70}$ dollars.