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. $\newline$ 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. $\newline$ $A=2000e^{0.1t}$ $\newline$ How long did it take Noah to pay off his loan if the amount he paid to his father was equal to $\$2450$ ? $\newline$ Give an exact answer expressed as a natural logarithm. $\newline$ $\square$

Full solution

Q. 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.

\newline

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.

\newline

A=2000e^{0.1t}

\newline

How long did it take Noah to pay off his loan if the amount he paid to his father was equal to

\$2450

\newline

Give an exact answer expressed as a natural logarithm.

\newline

\square

Set up equation: We have the equation $A = 2000e^{0.1t}$ and we know $A = \$2450$ . So we set up the equation $2450 = 2000e^{0.1t}$ .
Isolate $e^{0.1t}$ : Divide both sides by $2000$ to isolate $e^{0.1t}$ . We get rac{2450}{2000} = e^{0.1t}.
Simplify equation: Simplify $\frac{2450}{2000}$ to get $1.225 = e^{0.1t}$ .
Take natural logarithm: Take the natural logarithm of both sides to solve for $t$ . We get $\ln(1.225) = \ln(e^{0.1t})$ .
Apply logarithmic property: Using the property of logarithms that $\ln(e^x) = x$ , we can simplify the right side to $0.1t = \ln(1.225)$ .
Solve for $t$ : Divide both sides by $0.1$ to solve for $t$ . We get $t = \frac{\ln(1.225)}{0.1}$ .
Calculate final result: Calculate $t$ using a calculator. $t = \frac{\ln(1.225)}{0.1}$ .