document.write( "Question 1149061: Abby opened a retirement account with 9% APR and initial deposit of $8,000 compounded monthly.
\n" ); document.write( "(a) Find the exponential function that models the value of Abby's retirement account after t years.
\n" ); document.write( "(b) How long will it take to double the initial value? Estimate your answer to the nearest year.
\n" ); document.write( "(c) How long will it take to triple the initial value? Estimate your answer to the nearest year.
\n" ); document.write( "(d) How long will it take for the value of the account to reach 7 times the value of the account after 2 years? \n" ); document.write( "

Algebra.Com's Answer #853965 by MathTherapy(10801) $\"\"$ $\"About$

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document.write( "Abby opened a retirement account with 9% APR and initial deposit of $8,000 compounded monthly.\r\n" );
document.write( "(a) Find the exponential function that models the value of Abby's retirement account after t years.\r\n" );
document.write( "(b) How long will it take to double the initial value? Estimate your answer to the nearest year.\r\n" );
document.write( "(c) How long will it take to triple the initial value? Estimate your answer to the nearest year.\r\n" );
document.write( "(d) How long will it take for the value of the account to reach 7 times the value of the account after 2 years?\r\n" );
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document.write( "(a) Find the exponential function that models the value of Abby's retirement account after t years.\r\n" );
document.write( "Future-value-of-$1 formula: , with  = Future Value (Unknown, in this case)\r\n" );
document.write( "                                                                                         = Principal/Initial Deposit ($8,000, in this case)\r\n" );
document.write( "                                                                                         = Interest rate, as a decimal (9%, or .09, in this case)\r\n" );
document.write( "                                                                                         = Number of ANNUAL compounding periods (monthly, or 12, in this case)\r\n" );
document.write( "                                                                                         = Time Principal/Initial Deposit has been invested, in YEARS (t, in this case)\r\n" );
document.write( "                                                                                          \r\n" );
document.write( "                                                                                         ----- Substituting $8,000 for P, .09 for i, and 12 for m \r\n" );
document.write( "                                                                                         <=== Exponential function that models Abby's account-value, after t years\r\n" );
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document.write( "(b) How long will it take to double the initial value? Estimate your answer to the nearest year.\r\n" );
document.write( "                                                                                \r\n" );
document.write( "                                                                     ----- Substituting $16,000 for A, $8,000 for P, .09 for i, and 12 for m \r\n" );
document.write( "                                                                     \r\n" );
document.write( "                                                                               \r\n" );
document.write( "  \r\n" );
document.write( "                                                                            ----- Converting to LOGARITHMIC form\r\n" );
document.write( "    Time it'll take to double the initial value, or  = 7.730480505, or approximately 8 years.\r\n" );
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document.write( "(c) How long will it take to triple the initial value? Estimate your answer to the nearest year. \r\n" );
document.write( "                                                                          \r\n" );
document.write( "                                                               ----- Substituting $24,000 for A, $8,000 for P, .09 for i, and 12 for m \r\n" );
document.write( "                                                                \r\n" );
document.write( "                                                                         \r\n" );
document.write( "                                                                         \r\n" );
document.write( "                                                                      ----- Converting to LOGARITHMIC form\r\n" );
document.write( "Time it'll take to triple the initial value, or  = 12.25252171, or approximately 13 years.\r\n" );
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document.write( "** Notice that although 12.25252171 rounds off to 12 (nearest integer), the $8,000 investment, at the 12-year\r\n" );
document.write( "     juncture, will be less than TRIPLE, or less than $24,000 (3 * $8,000). This is why it's necessary to ROUND UP\r\n" );
document.write( "     to year 13, at which time, the $8,000 initial deposit will surpass the triple-value, $24,000.  \r\n" );
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document.write( "How long will it take for the value of the account to reach 7 times the value of the account after 2 years?\r\n" );
document.write( "\r\n" );
document.write( " <=== Exponential function that models Abby's account-value, after t years\r\n" );
document.write( " ---- Substituting 2 for t \r\n" );
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document.write( "Value of account after 2 years, or A = 9571.308235, or approximately $9,572.\r\n" );
document.write( "Seven (7) times the 2-year account value = 7(9,572) = $67,004. Now, when will the initial deposit ($8,000) increase to $67,004?\r\n" );
document.write( "                                                                             \r\n" );
document.write( "                                                                   ----- Substituting $67,004 for A, $8,000 for P, .09 for i, and 12 for m \r\n" );
document.write( "                                                                   \r\n" );
document.write( "                                                                  \r\n" );
document.write( "\r\n" );
document.write( "                                                                         ----- Converting to LOGARITHMIC form\r\n" );
document.write( "                                                            It'll take  = 23.70300853, or approximately 24 years for the $8,000 \r\n" );
document.write( "                                                            initial-deposit to increase to $67,004 (7 times its 2-year value, or 7 times appr. $9,572).

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