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Numerade Educator

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Problem 41 Hard Difficulty

A car dealer sells a new car for $ \$ 18,000 $. He also offers to sell the same car for payments of $ \$ 375 $ per month for five years. What monthly interest rate is this dealer charging?
To solve this problem you will need to use the formular for the present value $ A $ of an annuity consisting of $ n $ equal payments of size $ R $ with interest rate $ i $ per time period:
$$ A = \dfrac{R}{i} [1 - (1 + i)^{-n}] $$
Replacing $ i $ by $ x $, show that
$$ 48x(1 + x)^{60} - (1 + x)^{60} + 1 = 0 $$
Use Newton's method to solve this equation.

Answer

$\approx 0.76286 \%$

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Video Transcript

41. I have a wolf. Are you hurt by one minus one? Plus I in the end with negative and power where a equals 18,000 r equals 3 75 and n equals five years Fortune 16 months. So now I'm plugging in my known quantities I have 18,000 equals 3 75 eyes with my ex one minus one plus x with negative way Multiply both sides by the Super Bowl of 3 35 over X. That would give me 48. Yes, equals one minus one over one plus x the city of power multiplying everything by one plus x to the 60 of power I get 48 It's one plus X with the 60th power moves one plus x to this power Buying this one. Bring everything to one side and I have boarding in X finds one plus x to the sixties Power mine It's one plus X with 60 of power plus one equal zero, which is my ex. And now I'm gonna take the derivative of that. No, my prime. I take the product mobile. I'm gonna get 48 finds one plus to the six of negative 60 of power. No positives. You have power Loves 48 x time 60 one plus X to the 59th Power minus six with one less eggs to the 59 hours. So my f fine. It's been a simple morning. A one plus X in the 60th. No. 28 baby one plus X with 59 power minus 60 one plus X to the 59 power. Now I'm gonna factor out right with the factor out at 12. One plus eggs to the 59th Power You can't leave me where Work one plus x plus 2 40 2 40 Thanks, minus five. Distributing that four. So I get a crime equals O on X with 59 hour combined My life terms Wait 2 40 or were X minus one. That is my f frying. Now I get to Newton's equals x So start with 1%. So I get X equals 0.1 except two Quite 008 due to go to except three equals, You know, row seven six, you know to exit four 76 two and exit five equals at six, which so six 286 is 0.76 286%. And that would be our monthly. Never want to figure that out. Compounded her year you were just multiply that fight both. No compounded monthly per year. Compounded monthly.