SOLUTION: a business invests $8,000 in a savings account for 2 years. at the beginning of teh second year, an additional $2500 is invested. At the end of the second year, the account balance

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Question 187174: a business invests $8,000 in a savings account for 2 years. at the beginning of teh second year, an additional $2500 is invested. At the end of the second year, the account balance is $ 11445. What was the annual interest rate?
So far I used the compounding interest formula where I = P(1 + r)^t
for the 1 st year's interest = (8000 + 8000r)
and for the second year's interest was (1st year's interest + [(8000+2500) + (8000r)(1 + r)^1] = total income after two years of 11445.
to set up my quadratic equation I combined like terms
8000 + 8000r + [(10500 + 8000r)(1 + r)] = 11445
8000 + 8000r + [10500 + 8000r + 10500r + 8000r^2] = 11445
combining terms 18500 + 26500r + 8000r^2 = 11445
or 8000r^2 + 26500r = 11445 - 18500 or -7055
8000r^2 + 26500r + 7055 = 0
at this point I get stuck because the numbers are too high in order to effectively use the quadratic formula to solve for r
I believe the answer is .05 or 5% annual interest rate..can you show me how you worked through the quadratic formula...
Thank you very much!
Kerry
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
A=P%281%2Br%29 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%281%2Br%29%5Et to A=P%281%2Br%29%5E1 or A=P%281%2Br%29


A=8000%281%2Br%29 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%2B8000r Distribute


So after one full year, you now have 8000%2B8000r dollars in the account.


Now because "an additional $2,500 is invested", this pushes up the amount to 8000%2B8000r%2B2500 (just add 2500 to the last expression).


Combine like terms to get 10500%2B8000r


So the new principal is P=10500%2B8000r (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%281%2Br%29 Go back to the original formula


11445=%2810500%2B8000r%29%281%2Br%29 Plug in A=11445 and P=10500%2B8000r


11445=10500%2B18500%2Ar%2B8000%2Ar%5E2 FOIL


0=10500%2B18500%2Ar%2B8000%2Ar%5E2-11445 Subtract 11445 from both sides.


0=8000%2Ar%5E2%2B18500%2Ar-945 Combine like terms.


8000%2Ar%5E2%2B18500%2Ar-945=0 Rearrange the equation.


Notice we have a quadratic equation in the form of ar%5E2%2Bbr%2Bc where a=8000, b=18500, and c=-945


Let's use the quadratic formula to solve for r


r+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


Plug in a=8000, b=18500, and c=-945


r+=+%28-18500+%2B-+sqrt%28+342250000-4%288000%29%28-945%29+%29%29%2F%282%288000%29%29 Square 18500 to get 342250000.


r+=+%28-18500+%2B-+sqrt%28+342250000--30240000+%29%29%2F%282%288000%29%29 Multiply 4%288000%29%28-945%29 to get -30240000


r+=+%28-18500+%2B-+sqrt%28+342250000%2B30240000+%29%29%2F%282%288000%29%29 Rewrite sqrt%28342250000--30240000%29 as sqrt%28342250000%2B30240000%29


r+=+%28-18500+%2B-+sqrt%28+372490000+%29%29%2F%282%288000%29%29 Add 342250000 to 30240000 to get 372490000


r+=+%28-18500+%2B-+sqrt%28+372490000+%29%29%2F%2816000%29 Multiply 2 and 8000 to get 16000.


r+=+%28-18500+%2B-+19300%29%2F%2816000%29 Take the square root of 372490000 to get 19300.


r+=+%28-18500+%2B+19300%29%2F%2816000%29 or r+=+%28-18500+-+19300%29%2F%2816000%29 Break up the expression.


r+=+%28800%29%2F%2816000%29 or r+=++%28-37800%29%2F%2816000%29 Combine like terms.


r+=+0.05 or r+=+-2.3625 Divide


So the possible 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%2A0.05=5.


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Answer:

So the annual interest rate is 5%


That's quite a guess :)