SOLUTION: A book collector bought a first edition of " The Picture of Dorian Grey" by Oscar Wilde at auction for $1500. This book increases in value at a rate of 13% per year.At the same auc

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: A book collector bought a first edition of " The Picture of Dorian Grey" by Oscar Wilde at auction for $1500. This book increases in value at a rate of 13% per year.At the same auc      Log On


   

Question 1181245: A book collector bought a first edition of " The Picture of Dorian Grey" by Oscar Wilde at auction for $1500. This book increases in value at a rate of 13% per year.At the same auction, the collector bought a copy of "Pride and Prejudice" by Jane Austen for $1800. This one increases in value at a rate of 9% per year. Create an exponential function for each situation and use the functions to algebraically determine the amount of time it would take for the twobooks to be equal in value.
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Answer by ikleyn(52781) About Me  (Show Source):
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A book collector bought a first edition of "The Picture of Dorian Grey" by Oscar Wilde at auction for $1500.
This book increases in value at a rate of 13% per year.
At the same auction, the collector bought a copy of "Pride and Prejudice" by Jane Austen for $1800.
This one increases in value at a rate of 9% per year.
Create an exponential function for each situation and use the functions to algebraically determine
the amount of time it would take for the twobooks to be equal in value.
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For the book "The Picture of Dorian Grey" the formula for the exponential price growth is

    y = 1500%2A1.13%5Et,

where t is the time in years after starting moment.



For the book "Pride and Prejudice" the formula for the exponential price growth is

    y = 1800%2A1.09%5Et,

where t is the time in years after THE SAME starting moment.



To answer the problem question, we should solve this equation

    1500%2A1.13%5Et = 1800%2A1.09%5Et.



It is the same as

    1500%2F1800 = 1.09%5Et%2F1.13%5Et,

or

    0.833333 = %281.09%2F1.13%29%5Et

    0.833333 = 0.9646%5Et.



Take logarithm base 10 of both sides

    log(0.833333) = t*log(0.9646)
    
and find t

    t = log%28%280.833333%29%29%2Flog%28%280.9646%29%29%29 = 5.059 years.



ANSWER.  In 5.059 years, the two books will be at the same value.

Solved, answered and carefully explained.


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