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Naomi is going to invest $\$ 78,000$ and leave it in an account for 7 years. Assuming the interest is compounded daily, what interest rate, to the nearest hundredth of a percent, would be required in order for Naomi to end up with $\$ 91,000$ ?

Step 1 ：The problem is asking for the interest rate required for an investment of $78,000 to grow to $91,000 in 7 years with daily compounding.

Step 2 ：The formula for compound interest is: \(A = P (1 + r/n)^{nt}\) where: A = the amount of money accumulated after n years, including interest. P = principal amount (the initial amount of money) r = annual interest rate (in decimal) n = number of times that interest is compounded per year t = time the money is invested for in years

Step 3 ：We can rearrange the formula to solve for r: \(r = ((A/P)^{1/(nt)} - 1) * n\)

Step 4 ：We know that A = $91,000, P = $78,000, n = 365 (since interest is compounded daily), and t = 7 years. We can substitute these values into the formula to find the interest rate r.

Step 5 ：Substituting the values we get: \(r = ((91000/78000)^{1/(365*7)} - 1) * 365\)

Step 6 ：Solving the above expression we get the value of r as approximately 0.022022190014756582

Step 7 ：Converting this to percentage we get approximately 2.202219001475658%

Step 8 ：Final Answer: The interest rate required for Naomi to end up with $91,000 after 7 years is approximately \(\boxed{2.20\%}\)