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^{0.1t}. 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.

Accepted Solution

Answer:t=2 yearsStep-by-step explanation:Given in the question that;Amount borrowed=$2000Amount to be paid=$2450Equation  [tex]A=2000e^{0.1t}[/tex]Take e=2.718The substitute in equation[tex]A=2000e^{0.1t} \\\\2450=2000e^{0.1t}[/tex]Divide both sides by 2000[tex]\frac{2450}{2000} =\frac{2000e^{0.1t} }{2000} \\\\\\1.225=e^{0.1t}[/tex]Introduce natural log on both sides to eliminate e, Remember In e^x=x[tex]In1.225=Ine^{0.1t}\\ \\\\0.2029=0.1t[/tex]Divide both sides by 0.1 to get t[tex]\frac{0.2029}{0.1} =\frac{0.1t}{0.1} \\\\\\t=2.03[/tex]