Question 1111913
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Rachel Spender wants to invest $4000 in savings certificates which bear an interest rate of 7.25% compounded semi-anually. How long a time period should she choose in order to save an amount of $4700?
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Future-value-of-$1 formula: {{{A = P(1 + i/m)^(mt)}}}, with {{{A}}} = Future Value (Unknown, in this case)
                                                                                        {{{P}}} = Principal/Initial Deposit ($4,000, in this case)
                                                                                        {{{i}}} = Interest rate, as a decimal (7.25%, or .0725, in this case)
                                                                                        {{{m}}} = Number of ANNUAL compounding periods (semiannually, or 2, in this case)
                                                                                        {{{t}}} = Time Principal/Initial Deposit has been invested, in YEARS (t, in this case)

How long a time period should she choose in order to save an amount of $4700?

                                                                       {{{A = P(1 + i/m)^(mt)}}} 
                                                               {{{"4,700" = "4,000"(1 + .0725/2)^(2t)}}} ----- Substituting $4,700 for A, $4,000 for P, .0725 for i, and 2 for m 
                                                               {{{"4,700"/"4,000" = (1 + .0725/2)^(2t)}}}
                                                                     {{{47/40 = (1.03625)^(2t)}}}
                                                                     {{{2t = log ((1.03625), (47/40))}}} ----- Converting to LOGARITHMIC form
Time it'll take the $4,000 investment to increase to $4,700, or {{{t = highlight((log (1.03625, (47/40))/2))}}} = 2.26446601 years, which needs to be ROUNDED UP
                                                                                                                                                             to {{{2&1/2}}} years, or 2 years, 6 months.
 
** Notice that although 2.26446601 rounds off to about {{{2&1/4}}} years, the $4,000 investment, at the {{{2&1/4}}}-year juncture, will increase to about
$4,695.16 (< $4,700). This is why it's necessary to ROUND UP to year {{{2&1/2}}}, or 2.5 years (at the semi-annual point), at which time, the $4,000 
initial deposit will exceed $4,700 (about $4,779.50, to be exact).</font></font></font></b></pre>