You Are Going To Purchase A New Car But Be Responsible

Question

You Are Going To Purchase A New Car But Being A Responsible Consumer You are going to purchase a new car, but being a responsible consumer means doing a little bit of research first. First, you find the vehicle you are purchasing and its price. Vehicle: Chevy Volt Price: $39,145 Current interest rate: 3% Using the function A(t)=P(1+ r n ) nt , create the function that represents your new car loan that is compounded monthly. The principle will be the price of the vehicle you selected, not how much you are putting down. Being a smart financial planner, you want to figure out how many months it will be until your principal is paid down to $10,000.00. Solve for t and show all of your work. Note that t will be negative because the number of months will decrease the principal. Lastly, you decide to keep track of your loan four times a month instead of monthly. Solve for the adjusted interest rate.

Dr. Jack HW Helper · Accepted Answer

Introduction When purchasing a vehicle, understanding the mechanics of loans and interest calculations is crucial for responsible financial management. This paper discusses the application of compound interest formulas to a car loan, specifically focusing on modeling the loan repayment and adjusting the interest rate for more frequent compounding periods. Using the example of a Chevy Volt priced at $39,145 with an annual interest rate of 3%, the goal is to determine the time required to pay down the loan to $10,000 and to adjust for quarterly compounding. Understanding the Loan Model The primary formula used to model the loan is the compound interest function: $$ A(t) = P \left(1 + \frac{r}{n}\right)^{nt} $$ where: - $A(t)$ is the amount owed at time $t$, - $P$ is the principal or initial amount, - $r$ is the annual interest rate expressed as a decimal, - $n$ is the number of compounding periods per year, - $t$ is the time in years. For monthly compounding, $n = 12$. The initial principal, $P$, equals the vehicle's price, i.e., $39,145. The current annual interest rate is 3%, or 0.03 in decimal form. Formulating the Loan Function Substituting the known values into the formula gives: $$ A(t) = 39145 \left(1 + \frac{0.03}{12}\right)^{12t} $$ which simplifies to: $$ A(t) = 39145 \left(1 + 0.0025\right)^{12t} = 39145 \times 1.0025^{12t} $$ This function models the amount remaining on the loan after $t$ years when compounded monthly. Calculating the Time to Pay Down to $10,000 To find $t$ when the remaining amount is $\$10,000$: $$ 10000 = 39145 \times 1.0025^{12t} $$ Dividing both sides by 39145: $$ \frac{10000}{39145} = 1.0025^{12t} $$ Calculating the ratio: $$ 0.2555 \approx 1.0025^{12t} $$ Applying natural logarithm to both sides: $$ \ln(0.2555) = \ln(1.0025^{12t}) = 12t \times \ln(1.0025) $$ Calculating the logarithms: $$ \ln(0.2555) \approx -1.366 $$ $$ \ln(1.0025) \approx 0.002498 $$ Solving for $t$: \[ t = \frac{-1.366}{12 \times 0.002498}

You Are Going To Purchase A New Car But Be Responsible

You Are Going To Purchase A New Car But Being A Responsible Consumer

Paper For Above instruction

Introduction

Understanding the Loan Model

Formulating the Loan Function

Calculating the Time to Pay Down to $10,000

Adjusting for Quarterly Compounding

Conclusion

References