Amber invests $\$ 1500$ in an investment that continuously compounds at an annual rate of $7.5 \%$. How much money will Amber have in the account after 4 years?

Step 1 ：Given that Amber invests $1500 in an investment that continuously compounds at an annual rate of 7.5%. We are asked to find how much money will Amber have in the account after 4 years.

Step 2 ：We use the formula for continuous compounding, which is \(A = P * e^{rt}\), where:

Step 3 ：\(A\) is the amount of money accumulated after n years, including interest.

Step 4 ：\(P\) is the principal amount (the initial amount of money).

Step 5 ：\(r\) is the annual interest rate (in decimal).

Step 6 ：\(t\) is the time the money is invested for in years.

Step 7 ：In this case, \(P = \$1500\), \(r = 7.5\% = 0.075\), and \(t = 4\) years. We can substitute these values into the formula to find \(A\).

Step 8 ：Substituting the given values into the formula, we get \(A = 1500 * e^{(0.075*4)}\)

Step 9 ：Solving the above expression, we get \(A = 2024.7882113640048\)

Step 10 ：Rounding off the above value to two decimal places, we get \(A = 2024.79\)

Step 11 ：Final Answer: After 4 years, Amber will have approximately \(\boxed{2024.79}\) in the account.