Question

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.

Answer 1

To find out how long it took Noah to pay off his loan when the amount he paid to his father was $2450, you can set up the equation:

A = 2000e^(0.1t)

You want to find t when A is equal to $2450:

2450 = 2000e^(0.1t)

Now, you need to solve for t. Start by dividing both sides of the equation by 2000:

2450/2000 = e^(0.1t)

Now, take the natural logarithm (ln) of both sides to solve for t:

ln(2450/2000) = ln(e^(0.1t))

Using the property of logarithms that ln(e^x) = x, we can simplify further:

ln(2450/2000) = 0.1t

Now, solve for t:

0.1t = ln(2450/2000)

To isolate t, divide both sides by 0.1:

t = (1/0.1) * ln(2450/2000)

t = 10 * ln(2450/2000)

So, the exact answer for how long it took Noah to pay off his loan is:

t = 10 * ln(2450/2000)

Explanation: