Jack asked Jill to marry​ him, and she has accepted under one​ condition: Jack must buy her a new ​$320,000 ​Rolls-Royce Phantom. Jack currently has ​$56,610 that he may invest. He has found a mutual fund with an expected annual return of 5 percent in which he will place the money. How long will it take Jack to win​ Jill's hand in​ marriage? Ignore taxes and inflation. Question content area bottom Part 1 The number of years it will take for Jack to win​ Jill's hand in marriage is    enter your response here years. ​ (Round to one decimal​ place.)

1. We need to determine the time \( t \) it will take for Jack's investment to grow to $320,000. We can use the formula for compound interest:

A = P(1 + r)^t

where:

- \( A \) is the amount of money accumulated after \( t \) years, including interest.

- \( P \) is the principal amount (the initial amount of money).

- \( r \) is the annual interest rate (decimal).

- \( t \) is the time the money is invested for in years.

2. Plug the given values into the equation:

320,000 = 56,610(1 + 0.05)^t

3. Divide both sides by 56,610 to isolate the exponential term:

\frac{320,000}{56,610} = (1 + 0.05)^t

4. Simplify the fraction on the left-hand side:

5.651 = (1.05)^t

5. Take the natural logarithm (ln) of both sides to solve for \( t \):

\ln(5.651) = \ln((1.05)^t)

6. Using the power rule of logarithms (\( \ln(a^b) = b \cdot \ln(a) \)):

\ln(5.651) = t \cdot \ln(1.05)

7. Divide both sides by \( \ln(1.05) \) to solve for \( t \):

t = \frac{\ln(5.651)}{\ln(1.05)}

8. Calculate the values using a calculator:

t \approx \frac{1.731}{0.049}

t\approx35.5years