4.  Trey is taking out a loan to buy a new car for $35,000. It is a 6-year loan with an APR of 5.85%. What will the monthly payment be? Round to the nearest cent. Use Desmos and the monthly payment formula.


To calculate the monthly payment for Trey's car loan, we can use the **monthly payment formula for loans**, also known as the **amortization formula**:

\[
M = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1}
\]

Where:
- $ M $ = monthly payment
- $ P $ = loan amount (principal) = $35,000
- $ r $ = monthly interest rate = $\frac{\text{APR}}{12}$
- $ n $ = total number of payments = number of months

### Step-by-Step Calculation:

1. **Loan amount (P)**: $35,000
2. **APR (Annual Percentage Rate)**: 5.85% (as a decimal = 0.0585)
3. **Number of years**: 6 years
4. **Number of monthly payments (n)**: $6 \times 12 = 72$ months
5. **Monthly interest rate (r)**: 
   \[
   r = \frac{0.0585}{12} = 0.004875
   \]

Now, using the formula:

\[
M = \frac{35,000 \times 0.004875 \times (1 + 0.004875)^{72}}{(1 + 0.004875)^{72} - 1}
\]

I'll calculate this next.The monthly payment for Trey's car loan will be **$577.58**, rounded to the nearest cent.

Would you like further details on the calculation or have any questions?

Here are 5 related questions to deepen your understanding:
1. What would happen to the monthly payment if the loan term increased to 7 years?
2. How would a lower APR, such as 3.5%, affect the monthly payment?
3. What would the monthly payment be if the principal amount were $40,000 instead of $35,000?
4. How much total interest would Trey pay over the life of this loan?
5. How would the monthly payment change if Trey made a $5,000 down payment?

**Tip:** Increasing the loan term reduces monthly payments but increases the total interest paid.

Trey is taking out a loan to buy a new car for $35,000. It is a 6-year loan with an APR of 5.85%. What will the monthly payment be? Round to the nearest cent. Use Desmos and the monthly payment formula.

Find out how to calculate the monthly payment for a $35,000 car loan with a 5.85% APR over 6 years. This step-by-step guide uses the loan amortization formula to break down the process. Ideal for students in Grades 10-12 learning about finance and interest rates.

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