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## Homework Statement

A vehicle purchase for $32,000 depreciates at a rate of 75% every 6 years. Another vehicle purchased for $16,000 depreciates at a rate of 50% every 4 years. Create an exponential function for each situation, and use the functions to algebraically determine the amount of time it would take for the vehicles to be equal in value.

2. Homework Equations

[tex] A = P(1 \pm i )^{n} [/tex]

A is the future amount

P is the present/principal amount

i = interest rate per compounding period in decimal form

n = number of compounding periods

## The Attempt at a Solution

Algebraically:

[tex] A_{1} = A_{2} [/tex]

[tex] P_{1}(1-i_{1})^{n} = P_{2}(1-i_{2})^{n} [/tex]

[tex] \log P_{1} + n\log (1-i_{1}) = \log P_{2} + n\log (1-i_{2}) [/tex]

[tex] n\log (1-i_{1}) - n\log (1-i_{2}) = \log P_{2} - \log P_{1} [/tex]

[tex] n = \frac{ \log P_{2} - \log P_{1}}{\log (1-i_{1}) - \log (1-i_{2}) } [/tex]

My issue is I don't know how to use the 6 years and the 4 years. I've done it with semi-annually, annually, quarterly, etc, but this one deals with years > 1 . For instance, if this was semi, it would be (75% / 2) /100 , to get i.