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Noah borrows $2000 from his father and agrees to repay the loan and any interest determined by his father as soon as he has the money.
The relationship between the amount of money, A, in dollars that Noah owes his father (including interest), and the elapsed time, t, in years, is modeled by the following equation.

A=2000e^(0.1 t)
How long did it take Noah to pay off his loan if the amount he paid to his father was equal to 
$2450? 
Give an exact answer expressed as a natural logarithm.
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Noah borrows $2000\$2000 from his father and agrees to repay the loan and any interest determined by his father as soon as he has the money.\newline The relationship between the amount of money, AA, in dollars that Noah owes his father (including interest), and the elapsed time, tt, in years, is modeled by the following equation.\newlineA=2000e0.1tA=2000e^{0.1t}\newlineHow long did it take Noah to pay off his loan if the amount he paid to his father was equal to $2450\$2450?\newline Give an exact answer expressed as a natural logarithm.\newline\square

Full solution

Q. Noah borrows $2000\$2000 from his father and agrees to repay the loan and any interest determined by his father as soon as he has the money.\newline The relationship between the amount of money, AA, in dollars that Noah owes his father (including interest), and the elapsed time, tt, in years, is modeled by the following equation.\newlineA=2000e0.1tA=2000e^{0.1t}\newlineHow long did it take Noah to pay off his loan if the amount he paid to his father was equal to $2450\$2450?\newline Give an exact answer expressed as a natural logarithm.\newline\square
  1. Set up equation: We have the equation A=2000e0.1tA = 2000e^{0.1t} and we know A=$2450A = \$2450. So we set up the equation 2450=2000e0.1t2450 = 2000e^{0.1t}.
  2. Isolate e0.1te^{0.1t}: Divide both sides by 20002000 to isolate e0.1te^{0.1t}. We get rac{2450}{2000} = e^{0.1t}.
  3. Simplify equation: Simplify 24502000\frac{2450}{2000} to get 1.225=e0.1t1.225 = e^{0.1t}.
  4. Take natural logarithm: Take the natural logarithm of both sides to solve for tt. We get ln(1.225)=ln(e0.1t)\ln(1.225) = \ln(e^{0.1t}).
  5. Apply logarithmic property: Using the property of logarithms that ln(ex)=x\ln(e^x) = x, we can simplify the right side to 0.1t=ln(1.225)0.1t = \ln(1.225).
  6. Solve for tt: Divide both sides by 0.10.1 to solve for tt. We get t=ln(1.225)0.1t = \frac{\ln(1.225)}{0.1}.
  7. Calculate final result: Calculate tt using a calculator. t=ln(1.225)0.1t = \frac{\ln(1.225)}{0.1}.

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