Question

Question: 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.

Info:
Vehicle: Chevy Volt
Price: $39,145
Current interest rate: 3%

We had to use the function A(t)=P(1+r/n)^nt and I already created the function that represents the new car loan being compounded monthly which is A(t)= 39,145(1+0.03/12)^12t So how do I figure out how many months it will be and solve for t?

Answer 1

\[\frac{ 10000 }{ 39145 }=(1+\frac{ .03 }{ 12 })^{12t}\]
\[\log _{(1+\frac{ .03 }{ 12 })}(\frac{ 10000 }{ 39145 })\frac{ 1 }{ 12}=t\]

Answer 2

you can use a change of base formula to compute the logarithm
\[\log _{1+\frac{ .03 }{ 12 }}(\frac{ 10000 }{ 39145 })=\frac{ \ln(\frac{ 1000 }{ 39145 }) }{ \ln(1+\frac{ .03 }{ 12 }) }\]

Answer 3

so is t = -1468.7? @VeritasVosLiberabit

Answer 4

Not quite check your math

Answer 5

by the way there is a typo in my change of base equation. Should be 10,000 not 1,000

Answer 6

so -546.56 then?

Answer 7

@VeritasVosLiberabit

Answer 8

now divide by 12 and that is t

Answer 9

and take the absolute value and that will be be number of months.