Question 187174
{{{A=P(1+r)}}} Start with the interest formula. 



Note: since we're only looking at intervals of a year, this means that the value of "t" is 1 and simplifies {{{A=P(1+r)^t}}} to {{{A=P(1+r)^1}}} or {{{A=P(1+r)}}}



{{{A=8000(1+r)}}} Plug in {{{P=8000}}} (the initial investment) and {{{t=1}}} (since we want to find the amount at the end of the first year)



{{{A=8000+8000r}}} Distribute



So after one full year, you now have {{{8000+8000r}}} dollars in the account.



Now because "an additional $2,500 is invested", this pushes up the amount to {{{8000+8000r+2500}}} (just add 2500 to the last expression). 



Combine like terms to get {{{10500+8000r}}} 



So the new principal is {{{P=10500+8000r}}} (this principal will be invested in the account for year #2)



Since after two years you have $11,445, this means that {{{A=11445}}}



{{{A=P(1+r)}}} Go back to the original formula



{{{11445=(10500+8000r)(1+r)}}} Plug in {{{A=11445}}} and {{{P=10500+8000r}}}



{{{11445=10500+18500*r+8000*r^2}}} FOIL



{{{0=10500+18500*r+8000*r^2-11445}}} Subtract 11445 from both sides.



{{{0=8000*r^2+18500*r-945}}} Combine like terms.



{{{8000*r^2+18500*r-945=0}}} Rearrange the equation.



Notice we have a quadratic equation in the form of {{{ar^2+br+c}}} where {{{a=8000}}}, {{{b=18500}}}, and {{{c=-945}}}



Let's use the quadratic formula to solve for r



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



{{{r = (-(18500) +- sqrt( (18500)^2-4(8000)(-945) ))/(2(8000))}}} Plug in  {{{a=8000}}}, {{{b=18500}}}, and {{{c=-945}}}



{{{r = (-18500 +- sqrt( 342250000-4(8000)(-945) ))/(2(8000))}}} Square {{{18500}}} to get {{{342250000}}}. 



{{{r = (-18500 +- sqrt( 342250000--30240000 ))/(2(8000))}}} Multiply {{{4(8000)(-945)}}} to get {{{-30240000}}}



{{{r = (-18500 +- sqrt( 342250000+30240000 ))/(2(8000))}}} Rewrite {{{sqrt(342250000--30240000)}}} as {{{sqrt(342250000+30240000)}}}



{{{r = (-18500 +- sqrt( 372490000 ))/(2(8000))}}} Add {{{342250000}}} to {{{30240000}}} to get {{{372490000}}}



{{{r = (-18500 +- sqrt( 372490000 ))/(16000)}}} Multiply {{{2}}} and {{{8000}}} to get {{{16000}}}. 



{{{r = (-18500 +- 19300)/(16000)}}} Take the square root of {{{372490000}}} to get {{{19300}}}. 



{{{r = (-18500 + 19300)/(16000)}}} or {{{r = (-18500 - 19300)/(16000)}}} Break up the expression. 



{{{r = (800)/(16000)}}} or {{{r =  (-37800)/(16000)}}} Combine like terms. 



{{{r = 0.05}}} or {{{r = -2.3625}}} Divide 



So the <i>possible</i> answers are {{{r = 0.05}}} or {{{r = -2.3625}}}

  
  
However, since you CANNOT have a negative interest rate, this means that we can ignore {{{r = -2.3625}}}.



So this means that the only answer is {{{r = 0.05}}}



Now multiply by 100 (to convert to a percentage) to get {{{100*0.05=5}}}. 



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Answer:


So the annual interest rate is 5%



That's quite a guess :)