Question 155377
# 2


{{{A=P(1+r)}}} Start with the given formula
 
 
{{{A=10000(1+r)}}} Plug in {{{P=10000}}}
 

 
{{{A=10000+10000r}}} Distribute
 


So at the end of the first year, he has {{{10000+10000r}}} dollars in the account
 


Since "At the beginning of the second year, an additional $3500 is invested", this means that we simply add 3,500 to the amount {{{10000+10000r}}} to get {{{10000+10000r+3500=13500+10000r}}}
 


So at the beginning of the second year, he invests {{{13500+10000r}}} dollars
 
 
So this time {{{P=13500+10000r}}}
 
 
{{{A=P(1+r)}}} Go back to the given formula
 

 
{{{15569.75=(13500+10000r)(1+r)}}} Plug in {{{A=15569.75}}} (this is the amount that is in the account after the second year) and {{{P=13500+10000r}}}
 

 
{{{15569.75=13500+13500r+10000r+10000r^2}}} FOIL



{{{0=13500+13500r+10000r+10000r^2-15569.75}}} Subtract 15,569.75 from both sides
 

 
{{{A=10000r^2+23500r-2069.75}}} Combine like terms



{{{A=1000000r^2+2350000r-206975}}}  Multiply <b>every</b> term by the 100 to clear the decimals.



Notice we have a quadratic equation in the form of {{{ar^2+br+c}}} where {{{a=1000000}}}, {{{b=2350000}}}, and {{{c=-206975}}}



Let's use the quadratic formula to solve for r



{{{r = (-b +- sqrt( b^2-4ac ))/(2a)}}} Start with the quadratic formula



{{{r = (-(2350000) +- sqrt( (2350000)^2-4(1000000)(-206975) ))/(2(1000000))}}} Plug in  {{{a=1000000}}}, {{{b=2350000}}}, and {{{c=-206975}}}



{{{r = (-2350000 +- sqrt( 5522500000000-4(1000000)(-206975) ))/(2(1000000))}}} Square {{{2350000}}} to get {{{5522500000000}}}. 



{{{r = (-2350000 +- sqrt( 5522500000000--827900000000 ))/(2(1000000))}}} Multiply {{{4(1000000)(-206975)}}} to get {{{-827900000000}}}



{{{r = (-2350000 +- sqrt( 5522500000000+827900000000 ))/(2(1000000))}}} Rewrite {{{sqrt(5522500000000--827900000000)}}} as {{{sqrt(5522500000000+827900000000)}}}



{{{r = (-2350000 +- sqrt( 6350400000000 ))/(2(1000000))}}} Add {{{5522500000000}}} to {{{827900000000}}} to get {{{6350400000000}}}



{{{r = (-2350000 +- sqrt( 6350400000000 ))/(2000000)}}} Multiply {{{2}}} and {{{1000000}}} to get {{{2000000}}}. 



{{{r = (-2350000 +- 2520000)/(2000000)}}} Take the square root of {{{6350400000000}}} to get {{{2520000}}}. 



{{{r = (-2350000 + 2520000)/(2000000)}}} or {{{r = (-2350000 - 2520000)/(2000000)}}} Break up the expression. 



{{{r = (170000)/(2000000)}}} or {{{r =  (-4870000)/(2000000)}}} Combine like terms. 



{{{r = 17/200}}} or {{{r = -487/200}}} Simplify. 



So the possible answers are {{{r = 17/200}}} or {{{r = -487/200}}} 

  
which approximate to {{{r=0.085}}} or {{{r=-2.435}}}



However, since a negative interest rate doesn't make much sense, this means that the only solution is {{{r=0.085}}} which is the percentage 8.5%



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Answer:

So the interest rate is 8.5%